A note on conductors of Frey representations at $2$

In 2000, Darmon introduced the notion of Frey representations within the framework of the modular method for studying the generalized Fermat equation. A central step in this program is the computation of their conductors, with the case at the prime $2$ presenting particular challenges. In this article we study the conductor exponent at $2$ for Frey representations of signatures $(p,p,r)$, $(r,r,p)$, $(2,r,p)$, and $(3,5,p)$, all of which have hyperelliptic realizations. In particular we are able to determine the conductor at $2$ for even degree Frey representations of signature $(p,p,r)$ and $(3,5,p)$ and all rational parameters.