I think it is the characteristic. $A_5$ naturally sits in $\text{SO}(3)$ as the group of rotational symmetries of the icosahedron. Pulling $A_5$ back along the double cover $\text{SU}(2) \to \text{SO}(3)$ gives a double cover of $A_5$, the binary icosahedral group $G$, and this group is therefore equipped with a $2$-dimensional complex representation.
By this MO question, this representation is defined over $\mathcal{O}_K$ for a number field $K$, so we can reduce it $\bmod 2$; the double cover $G \to A_5$ has kernel $\{ \pm I \}$ which is killed when working $\bmod 2$, so we get a $2$-dimensional representation of $A_5$ in characteristic $2$.