This question is vague on purpose but I hope it should make enough sense.
I don't think it matters which type of objects I'm working with, but just in case it does, I'll point out that I'm working with Lie algebras.
Let $A$ and $B$ be "algebraic objects" (but maybe not abstract groups?). Suppose $\varphi: A \to B$ is a morphism and $C \subset \ker \varphi$ is an ideal (whatever that means in this setting). Of course we get an induced map $\tilde{\varphi} : A/\ker \varphi \to B$, but do we also have some sort of induced map $\hat{\varphi} : A/C \to B$?
This seems to be a standard trick in the literature but I can't seem to find any sources or explanations....
Let $K$ be the kernel of $\varphi$. Then by the third isomorphism theorem we have $$A/K\cong (A/C)/(K/C)$$ We use this to construct $\hat{\varphi}:A/C\to B$ with kernel $K/C$, factoring through the surjection $A/C\to A/K$ and composing with $\tilde{\varphi}$.