We consider a group finite $G$. We say a set $A\subset G$ injects a set $B$ if $|A+B| = |A||B|$, and let $I(B) = \max \{|A| :A\text{ injects } B\}$.
For a subgroup $H$, it is well-known that $I(H) = |G|/|H|$. Now, if we consider $H' = H \setminus \{e\}$, we still have that $I(H') = I(H)$, if $|H| > 2$, as then we can only fit 1 copy of $H'$ into each coset of $H$.
However, given a second subgroup, $F$, $I(F+H')$ is not always equal to $I(F+H)$.
Letting $G$ be the symmetric group on $n$ symbols, we have cases where $I(F+H') = |G|/|F+H'|$. For example, letting $\sigma = (1,2,3\dots n), \tau = (1,2)$, this occurs when $F$ is generated by $\sigma^{-1}\tau$, and $H$ is generated by $\sigma$. (this can be derived from this paper of Aaron Williams)
In my research I'm hoping to establish upper bounds of $I(F+H')$ for particular $G, F$ and $H$. Is there some literature about such problems? Are techniques of Ruzsa Calculus well-suited, or is some other topic in additive combinatorics more related?
