Let $G$ be a $p$-group and $H$ be a subgroup of $G$. Set $V$ be a pretransfer map from $G$ to $H$.
That is, $$V(g)=\prod_{t\in T} tg(t.g)^{-1}=\prod_{t\in T_0}tg^n_tt^{-1} $$
Can we say that the image $V(G)$ is contained in a (special subgroup) of $H$? Possible under some assumption on the structure of $H$. For example if $H$ is a regular $p$-group what can we say about the image $V(G)$?
