Asymptotic Growth of Trivial Summands in Tensor Powers

Given a finite-dimensional representation $V$ over an algebraically closed field of an abstract group $G$, we consider the number of the trivial summand counted with multiplicity in the direct sum decomposition of $V^{\otimes n}$. We give necessary and sufficient conditions when the field is of characteristic $0$ and when the field is of characteristic $p$ so that $(V^{\otimes n}{)}_n$ has a subsequence $(V^{\otimes n_k}{)}_k$ such that $V^{\otimes n_k}$ contains enough trivial summands when $k$ is sufficiently large.