A question on the effective cone Let $X$ be a projective variety and $G$ a finite group acting on $X$. We consider the quotient $\pi:X\rightarrow Y :=X/G$. 
I'm interested in the relation between $Eff(X)$ and $Eff(Y)$. In particular, is it true that if $Eff(X)$ has infinitely many extremal rays then $Eff(Y)$ has infinitely many extremal rays as well?
 A: No.  Let $X = E \times E$ with $E$ an elliptic curve and let $G = \mathbb Z_2 \oplus \mathbb Z_2$, with each factor acting on one of the $E$'s by the involution and fixing the other.  The quotient is $\mathbb P^1 \times \mathbb P^1$.  The effective cone of $E \times E$ is round, while the effective cone of $\mathbb P^1 \times \mathbb P^1$ is polyhedral.
The converse seems true, though: if $X$ has polyhedral effective cone, then so does $Y$, spanned by the pushforwards of the generators of the cone for $X$.
A: Here is an example with $Pic(X)$ finitely generated. Take $X = \overline{M}_{1,n}$ the moduli space of genus one curves with $n$ marked points. Then $Pic(\overline{M}_{1,n})$ is finitely generated.
Now, by Theorem $1.1$ of this paper:
https://www2.bc.edu/dawei-chen/Extremal.pdf
we have that $Eff(\overline{M}_{1,n})$ is not finitely generated. Now, consider the action of the symmetric group $S_n$ on $\overline{M}_{1,n}$ permuting the markings, and the quotient $\widetilde{M}_{1,n}:=\overline{M}_{1,n}/S_n$. Then, by Theorem $5.1$ of the same paper we get that $Eff(\widetilde{M}_{1,n})$ is the closed, simplicial cone generated by the boundary divisors.
Therefore, $Eff(\overline{M}_{1,n}/S_n)$ is finitely generated while $Eff(\overline{M}_{1,n})$ is not. 
