Take $C: y^2 = x^6 + b$.  There is a $(2,2)$-isogeny $J_C \to E_b^2$ corresponding to the factorization $\{x^2 - \beta_1, x^2 - \beta_2, x^2 - \beta_3\}$ of $x^6 + b$, where the $\beta_i$ are the roots of $x^3 + b$.  While these $\beta_i$ may only be defined over some extension of $\mathbb{F}_q$, the set of factors is rational over $\mathbb{F}_q$, and therefore the isogeny is defined over $\mathbb{F}_q$.

I think a good reference for this (over number fields) is Section 3 of Howe, Leprevost, and Poonen's [Large torsion subgroups of split Jacobians of curves of genus two or three](https://arxiv.org/abs/math/9809210).