Is there an explicit description of (the moduli space of) equivariant holomorphic embeddings from the 2-sphere $S^2$ to the quintic $Q$, such that $f_*[S^2]=2 \in H_2(Q,L;\mathbb{Z})?$

I'd like to know a phenomenology of what one can find, e.g., that only two or three kind of maps would have to be considered, for example the ones giving rise to an open surface, or to an unoriented surface with a crosscap.

Here the (Fermat) quintic is
$$ Q = \left\{ (x_1:x_2:x_3:x_4:x_5) \in \mathbb{CP}^4 \left| \quad \sum_{i=1}^5 x_i^5 = 0 \right.\right\},$$
the antiholomorphic involution is taken to be
$$\sigma: \mathbb{CP}^4 \ni (x_1:x_2:x_3:x_4:x_5) \mapsto (\overline{x}_2:\overline{x}_1:\overline{x}_4:\overline{x}_3:\overline{x}_5),$$
$L$ is the fixed locus of $Q$ under $\sigma$ and $\Omega: S^2 \to S^2$ is an antiholomorphic involution; equivariant means $f\circ\Omega=\sigma\circ f.$

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Perhaps, a simpler question is to first find all the (class of) embeddings, without reference to equivariance, and then check which ones are well-behaved with respect to equivariance. Being maps from $\mathbb{CP}^1$ into $Q$ they should be somewhat more rigid then the general genus $g$ case.

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I think another way of phrasing this is to provide a check for Clemens conjecture restricted to embeddings in Fermat quintic with $d=2$

Note. (Correct me if I'm wrong) Here $d$ can be thought more or less equivalently either as the homogeneous degree of the map or as the class in $H_2(Q;\mathbb{Z}).$