The purpose of this paper is to introduce and study the
notion of a vector-valued $\pi\text{-invariant}$ mean
associated to a unitary representation $\pi$ of a locally
compact group $G$ on $\mathcal{S},$ a self-adjoint linear
subspace containing $I$ of $\mathcal{B}(H_\pi).$ We obtain,
among other results, an extension theorem for
$\pi\text{-invariant}$ completely positive maps and
$\pi\text{-invariant}$ means which characterizes amenability
of $G.$ We also study vector-valued means on $\mathcal{S}$ of
$\pi\text{-(weakly)}$ almost periodic operators on $H_\pi.$