Let $X$ be a hyperelliptic curve, $\pi: X \to \mathbb{P}^{1}$ denote the ramified covering and $W$ the set of Weirstrass points. Let $F,G$ be two involution invariant (with respect to the hyperelliptic involution of $X$) bundles on $X$ fitting in the short exact sequences

$0 \to \pi^*E \to F \to \bigoplus_{w \in W} F^{⁻} \otimes \mathcal{O}_w \to 0$,

$0 \to \pi^*E' \to G \to \bigoplus_{w \in W} G^{⁻} \otimes \mathcal{O}_w \to 0$,

where $E,E'$ are parabolic vector bundles on $\mathbb{P}^{1}$ and $F^{-}, G^{-}$ denote the negative eigenspaces of the bundles $F,G$ (with respect to the involutions on $F$ and $G$ induced by the hyperelliptic involution of $X$).

Then does $\mathcal{E}nd(E) \simeq \mathcal{E}nd(E')$ imply that $\mathcal{E}nd(F) \simeq \mathcal{E}nd(G)$?

Any hints/suggestions will be helpful.