A Finite Skew Brace with Perfect Additive Group and Almost Simple Multiplicative Group

We construct a finite skew brace whose additive group is perfect and whose multiplicative group is non-perfect and almost simple. This gives an affirmative answer to Problem 20.109 in the twenty-first edition of the Kourovka Notebook. The additive and multiplicative groups of our example are $$\PSU_5(64)\times A_5 \quad\text{and}\quad \Aut(\PSU_5(64)),$$ respectively. The construction combines a skew brace with additive group $A_5$ and multiplicative group $(C_5\rtimes C_4)\times C_3$, the splitting of the automorphism extension of $\PSU_5(64)$, and a semidirect product of skew braces.