Stem extension (reviewed)

A short exact sequence of groups $$ 1 \to K \to C \to G \to 1 $$ is a stem extension if $K$ is contained in both the center and the commutator of $G$ (we also refer to $K$ as a stem subgroup of $C$). For fixed $G$, the size of $C$ is bounded. For any extension with $C$ of maximal size, the kernel $K$ is isomorphic to the Schur multiplier of $G$. In general, $C$ is only determined up to isoclinism in such an extension, but if $G$ is perfect then $C$ is determined up to isomorphism.