The PM in question wanted a rehash of something I posted about what types of M4 Shermans might be seen in US hands in Europe. Rather than find that somewhat limited summary, I decided to re-do with greater detail in my reply:
M4: The original design with radial petrol engine, M4A1 had the same engine but a cast hull, so I will deal with them together. Initially the preferred engine, both M4 and M4A1 (75) saw service from Torch to war's end. They would have been the sole combat models until late '43, early '44 and between the two would still just have been the largest percentage of M4's by VE day. Both later got better front hull armour (thicker at 47° vs older 56° from vertical). From early '44 the M4A1 got the 76mm while some of the M4 were 105mm armed. From memory of photos the 105mm M4 always had the later 47° hull whilst many of those late 75mm M4 had a cast front (aka composite hull). A composite hull M4 would be a M4A1 in ASL terms. The 76mm M4A1 first saw service in Operation Cobra. Not sure when the 105mm M4 saw combat, but I suspect late French campaign. The British fitted 17lbr to M4 but not to M4A1. What looks like a 17lbr M4A1 is actually a 17lbr M4 (composite hull).
M4A2: Diesel twin engines. Starting with 75mm and 56° hull, they progressively went to 47° hull and finally 76mm. Used by the US for training, saw US combat service only with the USMC. Most LL to the USSR, Britain and France, in order of priority, indeed one of the USMC batches was "stolen" from a USSR allocation. Only the USSR used the 76mm versions.
M4A3: Ford petrol twin engine. Had all the variations of the M4A2, but also had a 105mm version. Indeed it is quite difficult to tell the 2 apart, only the different horizontal engine decks are a good guide. Though some may have been used mid-Italy campaign, it was really D-Day onwards when they would have seen much service. 75mm, 47° hulls, I suspect Cobra+ and the 76mm and 105mm versions only becoming common by the end of the French campaign. The USMC eventually switched to the M4A3 from the M4A2. The US switched from considering the M4/M4A1 engine as the preferred one to the M4A3's Ford.
M4A3E2: Doled out in handfuls to (mainly) M4A3 battalions. I think they were preferentially given to separate tank battalions allocated to infantry divisions, though some saw use in armoured divisions. NW Europe only.
M4A4: 5 car engines fused together!!! US training only, LL to Britain and possibly France. Only 56° and 75mm, though the British fitted their 17lbr.
M4A5: Not a Sherman, but a type designation for the Canadian Ram tank.
M4A6: Diesel radial engine. Training/development only.
Italy: M4 & M4A1 throughout the campaign. M4A3 starting to appear mid-'44, likely as new battalions fed in.
NWE: Initially mainly M4 & M4A1 with some M4A3. In the immediate post invasion many separate and armoured division battalions were shipped straight from the US. These seem to be mainly M4A3 variants.
Mixing: M4 and M4A1 were practically interchangeable, so while many units would have started as pure M4 or M4A1, replacements could have been either. Naturally the USA preferred that M4/M4A1 were not mixed with M4A3 but there were times like very late '44 when the USA was running short due to losses. The British either offered or gave the USA some of their Shermans at that time. So you would see awkward mixes, especially when allocating 76mm and 105mm variants. The US could readily support logistically such less than optimal mixes.
Gun Mixes: From Cobra+ expect to see no more than 20% 76mm, the rest 75mm. By Bulge 40% 76mm and by VE 60%+ 76mm. The problem was that was not uniform. From memory, one of the post D-Day, direct from the US shipped armoured divisions came entirely equipped with M4A3(76) whilst most active M4A3 battalions could only dream of them.
105mm: Initially issued as a 3 tank platoon per tank battalion, later 1 was additionally added to each 17 tank M4-whatever company. So for 54 M4-? (75mm/76mm) tank battalion you could have 3 or 6 105mm M4/M4A3 in addition. They did not replace 75mm/76mm gun tanks. Production of M4A3(105) was roughly twice that of M4(105).
56° vs 47°. The early M4-? had 2" at 56°, later upgraded (except M4A4) to 2.5" at 47°. While almost the same effective horizontal line of sight thickness the 47° hull was a single plate without the driver/assistant driver hatch excrescences of the earlier 56° hull multi plate front, thus stronger. The 47° hulls also had bigger hatches allowing easier bail out.
As an addition to the above PM information I might as well complete the US battalion organisation by mentioning the light tanks and other lesser creatures.
The original OoB had a tank regiment with 1x3 company light tank battalion and 2x3 company medium battalions, each with 17 tanks/company and 3 in battalion HQ. The Battalion HQ had a 3 vehicle assault gun platoon and often a 3 vehicular 81mm halftrack MTR platoon. The assault gun platoon started with the likes of the T30 HMC (USVN 35), then the M8 HMC Scott (USVN 43) and finally by mid-late '44 the M4(105) or M4A3(105) (USVN 17). The MTR platoon used the M4, M4A1 and M21 halftracks. That organisation was in effect until mid-late '43 and indeed the 2nd, 3rd Armoured continued to use that "heavy" organisation until war's end. In practice light and medium companies were often swapped to give 3 equal battalions with 1 light and 2 medium companies. A US "heavy" Armoured Division had 6 light and 12 medium companies total in 2 regiments.
During '43 the heavy organisation was replaced by the light version. That abolished the tank regiment and instead had 3 tank battalions. Each battalion had a similar HQ and 1 light and 3 medium tank companies, like before with 17 or 18 tanks for a divisional total of 3 light and 9 medium companies. The upside was the infantry component got beefed up at battalion level.
The light tanks started with M3 and M3A1 but by Sicily they started to or had been replaced by M5 and M5A1. By Wacht am Rhein tiny numbers of M24 had appeared but took some time to displace the M5A1s.
Now be aware that all the above is just an overall broad sweep picture. For designing scenarios always use AARs, TO&E unit details where you can lay your hands on them. You will undoubtly find exceptions but the above should not lead you too far astray.