The dihedral group $D_{4}$ is generated by an element $a$ of order $4$, and an element $b$ of order $2$, satisfying the relation:

$(\ast)$ $ba = a^{n-1}b$

(i) Determine the number of group homomorphisms $f: D_{4} \rightarrow \mathbb{Z}_{4}$ with $f(b)=0$.



(ii) Determine the number of group homomorphisms $g: D_{7} \rightarrow \mathbb{Z}_{7}$ with $f(b)=3$.



Hint: Make use of the relation $(\ast)$.