Normal Subgroups and Quotient Groups · Hard problem

A homomorphism $\varphi: \mathbb Z_{12} \to D_4$ is determined by $h = \varphi(1) \in D_4$ (any choice works, since every element of $D_4$ has order dividing $4$, which divides $12$). For how many of the $8$ elements $h \in D_4$ does the resulting $\varphi$ have image of order exactly $2$?