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In this work we generalise the main result of arXiv:1812.05651 to the familyof hyperelliptic curves with potentially good reduction over a $p$-adic fieldwhich have degree $p$ and the largest possible image of inertia under the$\ell$-adic Galois representation associated to its Jacobian. We will provethat this Galois representation factors as the tensor product of an unramifiedcharacter and an irreducible representation of a finite group, which can beeither equal to the inertia image (in which case the representation is easilydetermined) or a $C_2$-extension of it. In this second case, there are twosuitable representations and we will describe the Galois action explicitly inorder to determine the correct one.

Wild Galois representations: a family of hyperelliptic curves with large inertia image