$\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, 
$$ \ord(a) \mid n$$
where $\ord(a)$ denotes the order of $a$ in the abelian group $G^\ab$. It follows that 
$$ \ord(a)\mid \gcd\nolimits_H[G:H]$$
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?