$A_4$ acts on a strip of 4 square tiles by exchanging them. The tiles are coloured with k-colours allowing repetitions. How many orbits of the action are there?
So I assumed that $A_4$ means an alternating group. I don't understand how alternating groups work and due to the question being a worded problem, how the set $X$ is structured.
Four colored tiles may be formalized as a $4$-tuple $(c_1,c_2,c_3,c_4)$ with $c_i$ a color drawn from a palette of $k$ colors, or equivalently a number between $1$ and $k$. This may in turn be viewed as a function $\{1,2,3,4\}\to \{1,\cdots,k\}$ on which $A_4$ acts by $(\sigma\cdot f)(x)=f(\sigma^{-1}x)$.
You could apply Polya's enumeration theorem, or work it out by hand. If you want to do it by hand, split into cases according to the multiplicities the colors. (E.g. four distinct colors, two colors on two tiles each, etc.)