How can I combine $C_2$ with $C_2$ to obtain $C_4$?

Question

Let $C_n$ be the cyclic group of order $n$. We know that $C_4$ is an extension of $C_2$ by $C_2$, so there must be a way of combining $C_2$ with itself in order to obtain $C_4$. But the only way I know of combining a group with another group is by way of the product operations, e.g. the direct, semi-direct, and free products. Yet, both the direct and the semi-direct product of $C_2$ by $C_2$ yield the Klein four group $V_4$, and if I'm not mistaken the free product will also not yield $C_4$ (though I'm less sure about this). Am I missing something?