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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;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 

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;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 

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

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;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 

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

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