Does a group homomorphism always produce a group extension. We all know the very famous group isomorphism theorems:
If $\varphi:G \rightarrow H$ is a group homomorphism them $G/\ker(\varphi)$ is isomorphic to $im(\varphi)$.
If $f: X \rightarrow Y$ is a covering map, we can think of $X$ as being a discrete fibre bundle over $Y$.
It seem as if a similar thing might happen with groups. There is a Wikipedia page for group extension https://en.wikipedia.org/wiki/Group_extension. Is this notion useful, can thinking about group homomorphism like this help? If so can you provide a good example where this is used, or some consequence of view the group this way?
 A: I am not sure what exactly you meant by saying "is this notion usefull" or what exactly you were adressing. However, in order to adress the question in your title:
Any group homomorphism $\varphi\colon G\to G'$ gives rise to a short exact sequence (canonical decomposition) $$0\to \ker \varphi\xrightarrow{i} G\xrightarrow{\varphi} \operatorname{im} \varphi\to 0$$
which is, by definition, precisely a group extension of $\operatorname{im}\varphi$ by $\ker \varphi$.
By the first isomorphism theorem this is (up to isomorphism) essentially $$0\to \ker \varphi\xrightarrow{i} G \xrightarrow{\pi} G\big/\ker\varphi\to 0$$ where $\pi$ denotes the canonical projection map.
In a similar manner you can always consider the direct product $G = N\times H$ of two groups $N,H$ together with the canonical inclusion $N\hookrightarrow N\times H$ and projection $N\times H\twoheadrightarrow H$ respectively, which provides $$0\to N\hookrightarrow N\times H \twoheadrightarrow H \to 0$$
Why is this usefull? For instance, suppose you are given a long exact sequence of $R$-modules $M_i$ $$\cdots \to M_i \xrightarrow{\phi_i} M_{i+1}\xrightarrow{\phi_{i+1}}M_{i+2}\to \cdots$$
the canonical decomposition allows you to extract short exact sequences by passing to the kernels and cokernels
$$0\to \operatorname{coker} \phi_{i-1}\to M_i \to \ker \phi_{i+1} \to 0$$
and this is used extensivelly in the algebraic machinery of algebraic topology whenever we are relating, for instance, the homology of a chain complex $(C_\bullet, \partial_\bullet)$ with the (co)homology of the dualized complex $(\operatorname{Hom}(C_\bullet),G))$ or similarly in the case of the complex $(C_\bullet,\partial_\bullet)$ after applying the $-\otimes G$ functor.
One particular result that immediately comes to my mind is the universal coefficient theorem for homology.
