# How many onto group homomorhisms are there from $\mathbb Z_2\times \mathbb Z_4$ onto $\mathbb Z_2\times \mathbb Z_4$? [duplicate]

I am trying find the number of onto group homomorphisms from $U(20)$ onto $U(15)$ where $U(n):=\{1\leq r\leq n: (r, n)=1\}$.

Here is what I tried.

Since $U(20)\cong \mathbb Z_4\times \mathbb Z_2\cong U(15)$, it follows that only we need to find the number of onto group homomorphisms from $\mathbb Z_2\times \mathbb Z_4$ onto $\mathbb Z_2\times \mathbb Z_4$. Let $f$ be one such onto group homomorphism. Then by first isomorphism theorem we must have $$\frac{\mathbb Z_2\times \mathbb Z_4}{\ker f}\cong \mathbb Z_2\times \mathbb Z_4.$$

This tells that $|\ker f|=1$ and hence $f$ is an isomorphisms.

NOW ?

How many isomorphisms are there from $\mathbb Z_2\times \mathbb Z_4$ to $\mathbb Z_2\times \mathbb Z_4$: is it a valid question ? If so, what should I reply ?

Can someone show me what I am doing wrong please ?