$\newcommand{\ab}{{\rm ab}} \newcommand{\ord}{{\rm ord}} $Let $G$ be a finite or profinite group. Consider the abelianized group $$G^\ab=G/G'$$ where $G'$ is the commutator subgroup of $G$.

Let $H\subset G$ be a subgroup of finite index. Consider the transfer (Verlagerung) map $$V_H\colon G^\ab\to H^\ab\,;$$ see https://en.wikipedia.org/wiki/Transfer_(group_theory)).

Fix $g\in G$ and write $a=gG'\in G^\ab$. I think that if $V_H(a)=0\in H^\ab$, then $na=0\in G^\ab$ where $n=[G:H]$, because ${\rm Cor}\circ{\rm Res}=n$. Therefore, $$ \ord(a) \,|\, n$$ where $\ord(a)$ denotes the order of $a$ in the abelian group $G^\ab$. It follows that $$ \ord(a)\,|\, \gcd\nolimits_H[G:H]$$ where $\gcd_H$ denotes the greatest common divisor over the subgroups of finite index $H$ such that $V_H(a)=0$.

Question. For given $a\in G^\ab$, what are the relations between the order $\ord(a)$ and the possible indices of the subgroups $H$ such that $V_H(a)=0$? In particular,

(1) Can $\ord(a)$ be strictly smaller than $\gcd_H [G:H]$ ?

(2) Do $\ord(a)$ and $\gcd_H[G:H]$ always have the same prime factors?