  U. Jezernik and P. Moravec have shown that if $G$ is a finite group with a
subgroup $H$ of index $n$, then nth power of the Bogomolov multiplier of $G$,
$\tilde{B_0}(G)^n$ is isomorphic to a subgroup of $\tilde{B_0}(H)$. In this
paper we want to prove a similar result for the center by center by $w$ variety
of groups, where $w$ is any outer commutator word.
