We study abstract finite groups with the property, called property $\hat{s}$,
that all of their subrepresentations have submultiplicative spectrum. Such
groups are necessarily nilpotent and we focus on $p$-groups. $p$-groups with
property $\hat{s}$ are regular. Hence, a 2-group has property $\hat{s}$ if and
only if it is commutative. For an odd prime $p$, all $p$-abelian groups have
property $\hat{s}$, in particular all groups of exponent $p$ have it. We show
that a 3-group or a metabelian $p$-group ($p \ge 5$) has property $\hat{s}$ if
and only if it is V-regular.