I don't have a reference, but here is a short argument.
Assume $G<H$ are compact groups (in our example take $H=\text{O}(n)$)
and recall that unitary representations of compact groups are completely reducible. Recall also that by Peter-Weyl every irreducible unitary representation is finite dimensional.

Assume first that $\rho$ is an irreducible finite dimensional continuous representation of $G$ and endow it with a $G$-invariant inner product. Fix $\pi$ to be any finite dimensional subrepresentation of the unitary induction $\text{Ind}_G^H(\rho)$, eg an irreducible subrepresentation.
Then, by Frobenius reciprocity, $\text{Hom}_G(\rho,\text{Res}_G^H \pi)\simeq \text{Hom}_H(\text{Ind}_G^H(\rho),\pi)$ and the latter is non-empty by our choice of $\pi$.
It follows that $\rho$ is a direct summand of $\text{Res}_G^H \pi$ by its irreducibility.

The case of a general $\rho$, continuous finite dimensional representation, follows easily.