For a semisimple complex Lie group G with a single outer involution \(\sigma \) and a hyperelliptic curve X on which \(\sigma \) acts as the hyperelliptic involution, the combination of the action of \(\sigma \) (as an outer involution) on principal G-bundles over X with the action of \(\sigma \) by pull-back gives an involution of the moduli space M(G) of principal G-bundles over X. Here, the notion of Galois G-bundle is used to describe fixed points of the mentioned involution. Specifically, the main original contribution of this work is to provide a proof that any Galois G-bundle that comes with a non-trivial finite-order automorphism commuting with the Galois structure (it will be such a fixed point in M(G)) admits a reduction of structure group to the centralizer of a non-central semisimple element of G. Results are also proved relating Galois G-bundles to Galois bundles whose structure group is the Langlands dual of G.