Some thoughts

let a be the automorphic number exceeding 9, then a*(a-1) is divisible by 100.

Therefore, a Xor a-1 is divisible by 2.

Therefore, a Xor a-1 is divisible by 4

Similarly

a or a-1 is divisible by 25.

Now a little trial and error:

a=25 works

49,50,51 don't divide by 4

a=76 works

let a be the automorphic number exceeding 99, then a*(a-1) is divisible by 1000.

Therefore, a Xor a-1 is divisible by 8.

Similarly

a or a-1 is divisible by 125.

A little more trial and error turns up two more automorphic numbers.