Manifolds modeled by an equivariant Hilbert cube

J. E. West posed the general problem of carrying over the basics of the theory of manifolds modeled by the Hilbert cube ($\equiv Q$-manifolds) into the equivariant realm. In particular, under the number 942 in 'Open problems in topology' he formulated the following problem: 'If $K$ is a locally compact $G$-CW complex, is the diagonal $G$-action on $X=K\times Q_G$ a $Q_G$-manifold? [$G$ is a compact Lie group and $Q_G=\prod_{i>0,\rho}D_{\rho,i}$ is the product of the unit balls of all the irreducible real representations of $G$, each representation disc being represented infinitely often.] What if $K$ is a locally compact $G$-ANR?' In this paper we construct a theory of $\mathbb Q$-manifolds for an arbitrary compact group $G$ in a scope that suffices for proving a characterization theorem for such manifolds.