A note on fixed points of automorphisms of infinite groups

Motivated by a celebrated theorem of Schur, we show that if Γ is a normal subgroup of the full automorphism group Aut(G) of a group G such that Inn(G) is contained in Γ and Aut(G)/Γ has no uncountable abelian subgroups of prime exponent, then [G, Γ] is finite, provided that the subgroup consisting of all elements of G fixed by Γ has finite index. Some applications of this result are also given.