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The following problem is proposed as Problem 18.57 in [The Kourovka Notebook,No. 18, 2014] by D. V. Lytkina: Let $G$ be a finite $2$-group generated byinvolutions in which $[x, u, u] = 1$ for every $x \in G$ and every involution$u \in G$. Is the derived length of $G$ bounded? The question is asked of anupper bound on the solvability length of finite $2$-groups generated byinvolutions in which every involution (not only the generators) is also left$2$-Engel. We negatively answer the question.

Non-solvable groups generated by involutions in which every involution is left $2$-Engel