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We define a variant of Benjamini-Schramm convergence for finite simplicialcomplexes with the action of a fixed finite group G which leads to the notionof random rooted simplicial G-complexes. For every random rooted simplicialG-complex we define a corresponding $\ell^2$-homology and the$\ell^2$-multiplicity of an irreducible representation of G in the homology.The $\ell^2$-multiplicities generalize the $\ell^2$-Betti numbers and we showthat they are continuous on the space of sofic random rooted simplicialG-complexes. In addition, we study induction of random rooted complexes anddiscuss the effect on $\ell^2$-multiplicities.

Equivariant Benjamini-Schramm Convergence of Simplicial Complexes and $\ell^2$-Multiplicities