Inclusion of varieties induces injective map of Baer invariants

Statement
Suppose $$\mathcal{V}_1$$ and $$\mathcal{V}_2$$ are subvarieties of the variety of groups with $$\mathcal{V}_1$$ contained in $$\mathcal{V}_2$$. Then, for any group $$G$$, there is a functorially induced injective map of Baer invariants:

$$\mathcal{V}_2M(G) \to \mathcal{V}_1M(G)$$

Related facts

 * Inclusion of varieties induces surjective map of second cohomology groups up to isologism