Let $G$ be a compact Lie group and $H$ a closed Lie subgroup. Let $i : H \hookrightarrow G$ denote the inclusion. Then given any finite-dimensional representation $\rho : H \rightarrow \mathrm{GL}(V)$ is it possible to find a finite-dimensional extension of $\rho$ to the whole group $G$, i.e. a finite-dimensional representation $\tilde{\rho} : G \rightarrow \mathrm{GL}(W)$ such that the following diagram commutes
$$\require{AMScd} \begin{CD} H @>{\rho}>> \mathrm{GL}(V)\\ @ViVV @VVjV \\ G @>{\tilde{\rho}}>> \mathrm{GL}(W) \end{CD}$$ where $j$ is injective.
$\textbf{Proposed solution:}$
Since we work with compact groups only, we can assume first that $\rho$ is irreducible. Then $\mathrm{ind}^G_H(\rho)$ defines a unitary representation of $G$, which we can write as a direct sum of finite-dimensional irreducible representations:
$\mathrm{ind}^G_H(\rho) = \bigoplus_{\alpha} \mu_{\alpha}$
The classical Frobenius reciprocity theorem states that given irreducible representations $\pi$ of $G$ and $\rho$ of $H$ we have that $\rho \subset \mathrm{res}^G_H(\pi)$ is a sub-representation if and only if $\pi \subset \mathrm{ind}^G_H(\rho)$ is contained in the induced representation.
Choosing now $\pi = \mu_{\alpha}$ we have by assumption: $\mu_{\alpha} \subset \mathrm{ind}^G_H(\rho)$ hence $\rho \subset \mathrm{res}^G_H(\mu_{\alpha})$. Therefore choose $\tilde{\rho} := \mathrm{res}^G_H(\mu_{\alpha})$ which is indeed finite-dimensional and the above diagram commutes.
If $\rho$ is not irreducible write it as a finite direct sum of irreducible ones, since $H$ is compact.