gr.group theory – Is a transfer homomorphism surjective?

Let $G$ be finite group with minimal number of generators$d$, and all his proper subgroups have at most $d-1$ as minimal number of generators.
Fix a normal subgroup $N$ of $G$.
For all subgroups $H$ of $G$ satisfying $[HN,N] subseteq N^2$; consider the transfer homomorphism $HN rightarrow H/H^2$.
Is it surjective?