Let $f:G\to H$ be a surjective morphism of groups and $f^{*}:[\textrm{ker}(f), G]\to [\{e_{H}\},H]$ defined by $f^{*}(L)=f(L)$. Show that $f^{*}$ takes finitely generated subgroups of $G$ into finitely generated subrgoups of $H$.
I was thinking about using the correspondence theorem (since $f^{*}$ is being used and so we know that $f^{*}$ is bijective). So if $L=\langle X\rangle \leq G$ with $X$ a finite subset then I want to prove that $f(L)$ is also finite generated. Could I say that $f(L)=\langle f(X)\rangle$? And since $f$ is surjective then $f(X)$ is finite and we finish.