M.R.Jones and J.Wiegold in [3] have shown that if $G$ is a finite group with a subgroup $H$ of finite index $n$ , then the $n$-th power of Schur multiplier of $G$ , $M(G)^n$ , is isomorphic to a subgroup of $M(H)$ . In this paper we prove a similar result for the centre by centre by $w$ variety of groups, where $w$ is any outer commutator word. Then using a result of M.R.R.Moghaddam [6], we will be able to deduce a result of Schur\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s type ( see [4,9] ) with respect to the variety of nilpotent groups of class at most $c$ $(c\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\geq 1)$, when $c+1$ is any prime number or $4$.

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