Let X be a hyperelliptic algebraic curve and let M(E6)\documentclass[12pt]{minimal}
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\begin{document}$$M(E_6)$$\end{document} be the moduli space of polystable principal E6\documentclass[12pt]{minimal}
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\begin{document}$$E_6$$\end{document}-bundles over X. Suppose, in addition, that the outer involution σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} of E6\documentclass[12pt]{minimal}
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\begin{document}$$E_6$$\end{document} acts as the hyperelliptic involution of X. Then, an automorphism of M(E6)\documentclass[12pt]{minimal}
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\begin{document}$$M(E_6)$$\end{document} is defined which acts by E↦σ∗(E∗)\documentclass[12pt]{minimal}
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\begin{document}$$E\mapsto \sigma ^*(E^*)$$\end{document}, where E is a principal E6\documentclass[12pt]{minimal}
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\begin{document}$$E_6$$\end{document}-bundle over X seen as a vector bundle through the fundamental irreducible 27-dimensional representation of E6\documentclass[12pt]{minimal}
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\begin{document}$$E_6$$\end{document}. In this paper, Galois E6\documentclass[12pt]{minimal}
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\begin{document}$$E_6$$\end{document}-bundles over X are defined and related to the fixed points of the above automorphism of M(E6)\documentclass[12pt]{minimal}
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\begin{document}$$M(E_6)$$\end{document}. If PE6\documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{P}\,}}E_6$$\end{document} is the centerless group with Lie algebra e6\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {e}}_6$$\end{document}, then Galois PE6\documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{P}\,}}E_6$$\end{document}-bundles over X are also defined and related to Galois E6\documentclass[12pt]{minimal}
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\begin{document}$$E_6$$\end{document}-bundles. Finally, a specific expression for a certain family of Galois E6\documentclass[12pt]{minimal}
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\begin{document}$$E_6$$\end{document}-bundles over X is given and some implications of the study in terms of representations of the fundamental group π1(X)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _1(X)$$\end{document} of the base curve are drawn.
