Large lifted product from a coset progression

Let $\pi :G\to G/[G,G]$ be the abelianization map and let $A$ be a finite $K$-approximate group. Suppose a coset progression $HP(x_1,\dots ,x_r;L_1,\dots ,L_r)$ lies in $\pi (A^4)$ and has size at least $\delta |\pi (A)|$. ThenA section of $\pi$ is multiplicative up to a bounded power of $A$ inside the commutator subgroup; successively separating the subgroup and progression coordinates proves the inclusion needed for this estimate.

Let $\pi :G\to G/N$ be a quotient group map, let $A$ be finite and symmetric, and suppose $P\subseteq \pi (A^m)$ has $|P|\ge \delta |\pi (A)|$. ThenChoose one lift in $A^m$ of each member of $P$ and multiply those lifts by $A^2\cap N$. Distinct fibers are disjoint, while $|\pi (A)||A^2\cap N|\ge |A|$.