  We define a variant of Benjamini-Schramm convergence for finite simplicial
complexes with the action of a fixed finite group G which leads to the notion
of random rooted simplicial G-complexes. For every random rooted simplicial
G-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 show
that they are continuous on the space of sofic random rooted simplicial
G-complexes. In addition, we study induction of random rooted complexes and
discuss the effect on $\ell^2$-multiplicities.
