Quotient of a proj variety by an involution

Question

Usually, if you have an affine variety defined by some equations and have an involution on it, it's quite easy to immediately see what the equations of the quotient of the variety by the involution should look like. For example, if $X: y^2 = f(x)$ and the involution is $\iota: (x,y) \mapsto (x,-y)$, then $X/\langle \iota \rangle$ has equation $y= f(x)$. For projective varieties, quotients are more difficult to make sense of, and one needs GIT. But say you have some "easy" projective variety $X = \mathbb{V}(ax^2 + bxy + cy^2 - z^2)\subseteq \mathbb{P}^2$, and you want to quotient this by the involution $\iota : [x:y:z] \mapsto [x:y:-z]$ (suppose for simplicity that this involution is fixed-point free). What are the explicit equations of $X/\langle \iota \rangle$? That is, is there a immediate way to see what the equations of the above quotient should look like?