Edit 2: This question does not make sense, as @Adam Z has brought me to understand that this quotient is not a subgroup. As @Shaun suggests, the problem is somewhat isolated, as my lack of prior experience brought me to conflate quotients and subgroups in the context of Puzzling Through Exact Sequences by Vakil.
In turn, @Adam Z kindly suggests familiarity with the isomorphism theorem before approaching Vakil's exposition.
For the diagram of homomorphisms given by: $$B \leftarrow A \rightarrow C$$
Does the quotient (which I assumed to be a subgroup) $$B/(\textrm{im}(\textrm{ker}(A \rightarrow B) \rightarrow C)) \ ... \ \textrm{incorrect}$$
Give the underlying set $S$ (allowing $B,A,C$ to denote the uderlying sets of the groups): $$ S ⊆ B-(A \ \cap \ C)$$
The answer is no, it gives—if I am not mistaken $$(B-A) \ \cup (B \ \cap \ A \ \cap \ C)$$
This question relates to Puzzling Through Exact Sequences by Ravi Vakil, something well beyond my current undergraduate courses.
Edit 1: Can somebody please explain the downvote?
Edit 3: Per the further advice of @Shaun on the context of this problem, I am preparing for a first course linear algebra this summer, and Vakil's exposition is a friendly introduction to exact sequences. Category theory is exciting to me, so the connections between sequences, groups, and linear algebra motivated me to ask this question. Vakil gives the homomorphisms and asks for the quotient specified above. I only have basic knowledge of group theory and short exact sequences from reading free course notes online. I thought I would try my hand at doing some problems but based on the reception of this question, I should wait until I take an actual course. I was looking for confirmation that I had interpreted Vakil's question correctly by bringing it into the context of set theory, which I have learned about from courses I've taken and also personal reading. Again, As @Adam Z kindly pointed out, I conflated groups and their underlying sets but also quotients and subgroups
Your question doesn't make sense since $\text{im}(\text{ker}(A\to B)\to C)$ is not a subgroup of $B$.
If you meant $B/\text{im}(\text{ker}(A\to C)\to B)$ in reference to the question given by Vakil, it will be the region represented by 3 and 4 in my labeled diagram from the document: labeled diagram
This is because 1+2 is $\text{ker}(A\to C)$ and so 2 is $\text{im}(\text{ker}(A\to C)\to B)$ (since taking the image of the map $A\to B$ is the same as quotienting by $\text{ker}(A\to B)$ and as long as you take the appropriate subgroup of the kernel, the same follows for restricted image). So 3+4 is $B$ quotiented by the region 2 which is what the question asks for.
When looking at the diagrams Vakil draws, it is very important to realize that these are not Venn diagrams of sets but instead ways to visualize subgroups and quotients of groups. For these to make sense, you have to be very comfortable with the isomorphism theorems of groups.