GL2(F2) group acting on an infinite element

Question

We are given a group $ GL_2(F_2)$ acting on the set $F_2 \cup \infty $ by fractional linear transformation $g•z=\frac{az+b}{cz+d}$. Here $F_2=\{0,1\} $. One of the elements of the group $ GL_2(F_2)$ would be matrix with a=1 b=0 c=1 d=1. Clearly ad-bc=1. If we take the action of this element on the element $\infty$ we get $\frac{\infty}{\infty}$. Which is not defined. So does this mean the group action is not defined. We add the $\infty$ element to the set to make sense of situation when we get $ \frac{1}{0}$