Diophantine equations involving powers of factorials

We are motivated by a result of Alzer and Luca who presented all the integer solutions to the relations $(k!{)}^n-k^n=(n!{)}^k-n^k$ and $(k!{)}^n+k^n=(n!{)}^k+n^k$. We consider the equations $(k!{)}^{n!}\pm k^n=(n!{)}^{k!}\pm n^k$ and $(k!{)}^n\pm k^{n!}=(n!{)}^k\pm n^{k!}$ and prove a similar statement.