Semi-Direct Product: Homomorphisms, Generators, Relations & Isomorphism Types

[mathusers]

thnx for helping on the previous post.

heres the next one, as usual any hints on how to approach the question would be greatly appreciated. i will attempt the questions with the hints :)

(1)
describe explicitly all homomorphisms

$\varphi : C_4 \rightarrow Aut(C_5)$

(2)
For each such homomorphism $\varphi$ describe the semidirect product $C_5 \rtimes_{\varphi} C_4$ in terms of generators and relations.

(3) How many distinct isomorphism types of groups of the form $C_5 \rtimes_{\varphi} C_4$ are there?

 

[morphism]

It will help to know what Aut(C_5) is. Hint: It's cyclic.