Why is it necessary for a quotient group to be defined for a normal *subgroup*.

Question

So I was thinking about the Second Isomorphism theorem, and noticed that quotient group $(A + B)/B$ could be written more nicely as $A/B$ if it were not required for the group representing the quotient group to contain $B$. In fact, if we just made it a condition that a subgroup $A$ were contained in the normalizer of a subgroup $B$, then we could define the set of cosets of $B$ under $A$ as a group under the same operation that quotient groups are usually afforded. My question then is, is there any context in which this is standard notation, and if not then is there either something wrong with my reasoning or is there just some sort of reason why it wouldn't be a very useful thing to consider?