So I am given an abelian group of order $8$ such that for all non-identity elements $x^2 = e$ (all elements have order two). So I know the answer is gonna be $168$, but I gotta prove this.
So far I have that there are exactly $7$ subgroups of order $4$, and each one has $6$ automorphisms. That gives me a total of $42$ automorphisms. Now there must be a justification for multiplying this number by $4$ to get the $168$. But I can't seem to find a justification for this.