The Kernel and Cokernel of a homomorphism (say of abelian groups) $\varphi : G \to H$ measure how $\varphi$ fails to be injective and surjective respectively. If the kernel is trivial, then $\varphi$ is injective, and dually for the cokernel. Let's focus on the kernel from here forwards, but I'm interested in related questions for the cokernel.
Checking if the kernel is trivial seems to throw away a lot of information, because we aren't making use of what the kernel actually is. Obviously we can use information about the kernel more economically by using the isomorphism theorems, but can we use it to minimally adjust $\varphi$?
Say $\varphi : G \to H$ has nontrivial kernel. Can we find a group $H'$ which is somehow closely related to $H$, with a hom $\varphi' : G \to H'$ which is closely related to $\varphi$, such that $\varphi'$ is injective? If so, can this "closely related"-ness be formalized as a universal property?
Edit:
Using $\varphi$ we can view $G$ as being "almost a subgroup" of $H$, where we might have to fudge some of the multiplication table to make $G$ fit. This fudging is measured by the kernel of $\varphi$. Say $\varphi$ isn't surjective. Then is there a way to extend $H$ in some minimal fashion in order to faithfully witness $G$ as a subgroup?
Edit 2:
To answer the "what have you tried" comment, let $C = H / Im (\varphi)$ be the cokernel of $\varphi$. Then $H' = C \times G$ is a new group which is similar to $H$ into which $G$ embeds. However, if there were relations between elements in the cokernel and elements in the image of $\varphi$, it feels like these may have been clobbered by this change. Is there a better construction which preserves more of the original structure of $H$?