I'm sure there must be lots of examples of three (infinite) groups $A$, $B$, $C$ where $A , $A\cong C$, $A\not\cong B$.
Here's one example with free groups: $\mathbf F(a^2,ab) < \mathbf F(a^2,ab,ab^{-1}) < \mathbf F(a,b)$, where A,C are a free group on 2 symbols and B is a free group on 3 symbols (in fact precisely the strings of even length in $\mathbf F(a,b)$.
But are there any examples that are just as easy but do not involve free groups (or equivalent constructions)?