Let $ G $ be a completely reducible algebraic group (in other words Maschke's theorem applies). Let $ \{ \rho_i \} $ be irreps of $ G $. Consider the direct sum of irreps $$ V=\bigoplus_i n_i (\rho_i) $$ where the $ n_i $ is the multiplicity of the irrep $ \rho_i $ in the direct sum. Let $ H $ be an algebraic subgroup of $ G $ such that the restriction of each $ \rho_i $ to $ H $ is still irreducible and moreover distinct $ \rho_i $ restrict to distinct irreps of $ H $ (this second condition is important otherwise you get counterexamples like the one given here https://math.stackexchange.com/a/4669169/758507). So the direct sum decomposition into $ H $ irreps is $$ V=\bigoplus_i n_i (\tau_i) $$ Here $ \tau_i $ denotes the $ H $ irrep corresponding to $ \rho_i $.
Prove that every $ H $ subrep of $ V $ must also be a $ G $ subrep of $ V $.
We are not assuming every subrep equivalent to one of the $\sigma_i$s is $G$-stable, only that the specific subreps corresponding to the $\sigma_i$s in some particular direct sum decomposition are. This seems to make things tricky; a subrep after all can be "diagonal" or "skew" WRT these $\sigma_i$s, to be made precise below.
I assume we're in a context where Maschke's theorem applies. Suppose $U_1,U_2,\cdots$ are the $G$-irreps. Every $G$-rep $V$ has a unique isotypical decomposition $V=V_1\oplus V_2\oplus\cdots$, where $V_i$ is a sum of $U_i$s. A $G$-subrep $W\le V$ also has an isotypical decomp $W=W_1\oplus W_2\oplus\cdots$, in which case $W_i\le V_i$ for each $i$.
Any isotypical rep $V_i$ is equivalent to $V_i\cong U_i\otimes\mathbb{F}^{m_{\large i}}$, where $\mathbb{F}$ the underlying scalar field, $m_{\large i}$ the multiplicity of $U_i$ in $V_i$, and $\mathbb{F}^{m_{\large i}}$ is a trivial rep. Every subrep of $U_i\otimes\mathbb{F}^{m_{\large i}}$ is of the form $U_i\otimes X_i$, where $X_i$ is any vector subspace of $\mathbb{F}^{m_{\large i}}$. (So for example, with $\mathbb{F}^2$, the diagonal line would be "skew" between the two copies of $U_i$ corresponding to the two coordinate axes in $\mathbb{F}^2$.)
WLOG then, $V=\bigoplus U_i\otimes\mathbb{F}^{m_{\large i}}$ is an isotypic decomposition, as both $G$-rep and $H$-rep, and thus any $H$-subrep is of the form $\bigoplus U_i\otimes X_i$, which is of course still a $G$-rep.