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In my course, if $f\colon M_2(\mathbb{Z}_2)\rightarrow M_2(\mathbb{Z}_2)$ is an automorphism then $$f( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix})=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ and $$f( \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix})=\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$ and $\forall A,B\in M_2(\mathbb{Z}_2)$ $$f(AB)=f(A)f(B)$$ and $$f(A+B)=f(A)+f(B)$$

I also know the orders of all the elements must be preserved and since the order of the ring is 16 they must all divide 16 (?) I also know that the only automorphism of $\mathbb Z_2$ is the identity, I'm not sure if I can use this in some way here. I'm not really sure how to proceed with this type of question other than exploring all the options for each element. I'm wondering if there is a more efficient way.

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    No, only the invertible elements have orders, and then you have to count the number of those to figure out the order of the group of units. – 2017-02-17

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