I had thought that during World War II they 1) only deployed "composite" squadrons and 2) never deployed with F6Fs. It turns out that I was wrong on both counts.
The FM-2 Wildcat was specifically developed for CVE-based VC squadrons. It was lighter and had a more powerful engine than the FM-1. The vertical fin and rudder were increased in size to compensate for the horsepower increase.
National Archives 80-G-224669
Grumman did a design study in mid-1943 of an F6F-3 modified for operation from CVEs. The "F6F-4" was to be armed with four .50 caliber guns instead of six and have the 75-gallon fuselage fuel tank deleted; the "F6F EX. W.T." was to have a greater wing span. Each wing tip was to be extended two feet and the ailerons moved outboard, with the span increase being split between the ailerons and the flaps by seven inches and 17 inches respectively. The study concluded that the F6F was faster but the FM-2 was smaller and had a lower stall speed and better takeoff performance, making them more suitable for operation from a CVE.
As it turned out, there appear to be many instances of CVEs deploying with Hellcats.
Barnes (CVE-20) reportedly deployed with VF-1 flying F6F Hellcats in November and December 1943. Some of the results were ugly. The pilot was killed in this landing attempt during workups in October. (Note the very unusual variation on the tri-color scheme.)
National Archives 80-G-202611This one wasn't fatal but still embarrassing for all concerned.
Some Sangamon-class CVEs deployments during World War II were with both a fighter squadron equipped with F6F Hellcats as well as a VT squadron assigned TBMs or a VC squadron operating SBDs and TBMs. The Sangamons (only four were built) were a bit longer and had a bigger hangar deck than the earlier U.S. Navy CVEs. The later Commencement Bay-class CVEs were similar in size and considered big enough and fast enough for Vought F4U Corsair and Grumman AF Guardian operation.