First way: Try to prove that the set
$\left\{\;\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}\;\;;\;\;\alpha,\beta,\gamma,\delta\in\{0,1\}\;\right\}$
is a complete set of representatives of the different cosets in $\,G/H\,$ (this implies that you also must prove these represent all the different elements in the quotient group). How many are there?
Second way: Define
$\,\phi: G\to K:=C_2\times C_2\times C_2\times C_2\,\,,\,\,\phi\begin{pmatrix}a&b\\c&d \end{pmatrix}:=\left(a,b,c,d\right)\pmod 2\in K\,$
Check the above is a group homomorphism...what is its kernel?