In base $2$ the solutions are [A306259](https://oeis.org/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](https://oeis.org/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;a=2;for(p=1,10^4,if(isprime(p),next);b=Mod(a,p^2)^((p-1)*p);if(b==1,print1(p,",");c++ ));c