= Solution
Let $\zeta=e^{2\pi i/7}$ and define
$$
\phi(x,y)=(\zeta^3x,\zeta y).
$$
Each term in the defining polynomial is multiplied by $\zeta^3$:
$$
(\zeta^3x)^3(\zeta y)+(\zeta y)^3+\zeta^3x
=\zeta^3(x^3y+y^3+x).
$$
Thus $\phi$ is a holomorphic automorphism of $X'$, with inverse obtained by replacing $\zeta$ by $\zeta^{-1}$. By the assumption it extends to a holomorphic automorphism of $X$; in the projective model it is
$$
[X:Y:Z]\longmapsto[\zeta^3X:\zeta Y:Z].
$$
It has order exactly seven because $\zeta$ is primitive and $y$ is not identically zero. Hence $\operatorname{Aut}(X)$ contains an element of order seven.
The extended coordinate function $\pi_x:X\to\mathbb C_\infty$ is a <meromorphic function as a holomorphic map to the Riemann sphere>. Compose it with the seventh-power map of the sphere and set
$$
\pi=\pi_x^7.
$$
This map is nonconstant and holomorphic as a map to $\mathbb C_\infty$. Since $\pi_x\circ\phi=\zeta^3\pi_x$,
$$
\boxed{\pi\circ\phi
=(\zeta^3\pi_x)^7
=\pi_x^7
=\pi}.
$$
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