Let $G$ be a solvable group and $f: G \rightarrow H$ an isomorphism. My question is if $H$ is a solvable group.
Let $1=G_0 \trianglelefteq G_1 \trianglelefteq ... \trianglelefteq G_n = G$ be a normal series of $G$ whose factors are abelian groups. Clearly, $1=f(G_0) \trianglelefteq f(G_1) \trianglelefteq ... \trianglelefteq f(G_n) = H$ is a normal series of $H$ but are the quotients abelian?
Is there a weaker condition for invariance?