# 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?