In base $2$ the solutions are A306259:
Composite numbers k such that $2^{(k(k-1))} \equiv 1 \pmod k^2$
From a comment: It contains all Fermat pseudoprimes to base 2, A001567.
The sequence starts:
21, 105, 165, 205, 231, 273, 301, 341, 385, 465
There are many solutions in other bases, the following pari/gp program computes them:
c=0;for(p=1,10^4,if(ispseudoprime(p),next);a=Mod(5,p^2)^((p-1)*p);if(a==1,print1(p,",");c++ ));c