 Isogenous elliptic curves can have different rational point groups

Elliptic curves isogenous over a finite field have the same number of rational points: the isogeny intertwines Frobenius, and multiplicativity of degree gives equal degrees for $1-\pi$. Their groups need not be isomorphic. Over ${\mathbb{F}}_7$, $y^2=x^3-x$ and $y^2=x^3+4x$ are two-isogenous and have groups $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/8\mathbb{Z}$, respectively.