Answer: 396

Small Hint:

An onto map from a finite set to itself is a permutation, and the condition says its sixth power is the identity

Big Hint:

Equivalently every cycle length divides $6,$ so subtract from $720$ the permutations that contain a $4$-cycle or a $5$-cycle

Solution:

A function from a finite set onto itself is a bijection, so $\pi$ is a permutation of six elements, and the condition says $\pi^6$ is the identity. A permutation satisfies $\pi^6 = \mathrm{id}$ exactly when every cycle in its cycle decomposition has length dividing $6.$ Among the possible lengths $1$ through $6,$ only $4$ and $5$ fail to divide $6.$

We subtract the permutations containing a $4$-cycle or a $5$-cycle from $6! = 720.$ Cycle type $4+1+1$ gives $\frac{6!}{4 \cdot 2!} = 90,$ type $4+2$ gives $\frac{6!}{4 \cdot 2} = 90,$ and type $5+1$ gives $\frac{6!}{5} = 144,$ for $90 + 90 + 144 = 324$ excluded permutations.

The count is $720 - 324 = 396.$