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U. Jezernik and P. Moravec have shown that if $G$ is a finite group with asubgroup $H$ 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 thispaper we want to prove a similar result for the center by center by $w$ varietyof groups, where $w$ is any outer commutator word.

Subgroup Theorems for the $\tilde{B_0}$-invariant of groups