Induction: prove $\,a^{2^n} \equiv 1\pmod{2^{n+2}}\,$ via congruences

Prove that:

$$a^{2^n} \equiv 1\pmod{2^{n+2}}$$

I have done the base case, and then have an assumption that:

$$a^{2^k} \equiv 1 \pmod {2^{k+2}}$$

I wish to prove the last step, that:

$$a^{2^{k+1}} \equiv 1 \pmod{2^{k+3}}$$

How would I do this?

First of all, you should specify that $a$ is odd, and $n > 0$.

And then, use the definition of the congruence relation $$ a^{2^k} \equiv 1 \pmod{2^{k+2}} $$ to write $$ a^{2^{k}} = 1 + c 2^{k+2}, $$ for some $c$, then take the square.

I'm assuming the you forgot to specify that $a$ is odd. From the induction hypothesis you know that $a^{2^{k}}=t 2^{k+2} + 1$ for some integer $t$. From this we get

$$a^{2^{k+1}}=(a^{2^{k}})^{2}=(t 2^{k+2} + 1)^2=t^2 2^{2k+4} + t2^{k+3} + 1$$

But $2^{2k+4} \equiv 0 \mod(2^{k+3})$ and so $$t^2 2^{2k+4} + t2^{k+3} + 1 \equiv 1 \mod(2^{k+3})$$