Even before the BMW 801D was completely debugged, the work on the improved 801E started. The goal was 2000 PS (take off). The supercharger's impeller received the 'Vorsatzlaufer', ie. the additional set of blades right before the 'main' impeller. How those additional blades looked like can be seen
here, though that is for the Jumo 213. The impeller was also further refined, per v. Gersdorff at al.
The reinforcements were incorporated in pistons, cylinder heads (being cast), reduction gear, while the oil system received the de-aerator centrifuge. Also the gearing for the 1st supercharger gear was increased. The result was the increase of 'normal' greatest boost at 1.62 ata, vs. 1.42 ata in the fully rated BMW 801D, on about the same altitude. The bench tests gave 2000 PS for take off, and 1710 PS at 5650 m. The time for the Notleistung was also increased from 3 (for 801D) to 5 minutes, and the engine was tested on the bench for up to 2300 PS. My guess is that those values are without the power needed for the cooling fan.
All the required changes were eventually the undoing for the BMW 801E. Basically, nothing from the 801D, the engine in volume production, could not be used on the 801E. Only about bout 80 were produced.
The allied post-war table, taken from the AEHS, lumps the 801E, F and S together:
The BMW 801S was based on the 801D, while incorporating some changes from the 801E: supercharger system, cylinder heads, increased 1st speed S/C gear ratio. The power achieved was on par with the 801E, but the engine was also running pretty much too late to matter - service use from late 1944 on. Increase of the boost up to 1.82 ata was to give 2200 PS for take off, again that would be without the power needed by the fan. A thread about the 801S:
here.
An offspring from the 801E was to be the 801F. Changes involved were: strengthening of the crankshaft, counterbalances with dampers for the crankshaft, bigger valves, different valve timing, bigger fuel injection pump, refined outer air intake starting from wing root, further increase of S/C gear ratios, improved internal and external aerodynamics of the 801TF power egg. The take off power was to be 2400 PS, the 'Kampflesitung' 2200 PS at sea level. During the bench testing 2600 PS was achieved, but the whole program was too late to matter.
Sometimes there is an argument whether the 801F ever saw service use, I always claiming it did not. For honesty sake, there are some documents around that state the Fw-190A-9 was outfitted with it. Mostly, the engine is noted in the same time being the 801F
and 801TS - the 'TS' being the power egg package of the 801S.
added: the web page talking about the BMW engines, among other. In Russian, but most of the stuff can be translated:
link