Small doubling in $m$-Engel groups

We study some inverse problems of small doubling type in the class of $m$-Engel groups‎. ‎In particular we investigate the structure of a finite subset $S$ of a torsion-free $m$-Engel group if $|S^۲| = ۲|S|+b$‎, ‎where $۰ \leq b \leq |S|-۴$‎, ‎for some values of $b$‎.