I'm trying to understand the Eichler-Shimura congruence which relates the Hecke operator $T_p$ to Frobenius at $p$ in characteristic $p$.

Two possible ways to compute $T_p$ mod $p$ seem to be:

A) Look at the map $Div(X_0(N)) \to Div(X_0(N))$ induced by the correspondence $X_0(N) \leftarrow X_0(Np) \to X_0(N)$ where the first map forgets the subgroup at $p$ and the second mods out by it. And then just take this map and look at the induced map $Div(X_0(N)_{F_p}) \to Div(X_0(N)_{F_p})$ in characteristic $p$.

B) Look directly at the correspondence $X_0(N)_{F_p} \leftarrow X_0(Np)_{F_p} \to X_0(N)_{F_p}$ in characteristic $p$ and try to compute directly the induced map $Div(X_0(N)_{F_p}) \to Div(X_0(N)_{F_p})$.

In case (A), starting with a point in $X_0(N)$, it has $p+1$ lifts to $X_0(Np)$ and thus the induced map on divisors will result in divisors of degree $p+1$. Explicitly, I'm getting: $$ (E,C) \to (E,(C+\ker(F))/\ker(F)) + p (E,(C+\ker(V))/\ker(V)) $$ which looks at lot like the standard Eichler-Shimura relation.

But in case (B), starting with an *ordinary* point in $X_0(N)_{F_p}$, it has $2$ lifts to $X_0(Np)$ --- one where we pick $\ker(F)$ as our group scheme of order $p$, and another for $\ker(V)$. Thus the induced map on divisors will result in divisors of degree $2$, and it seems to yield
$$
(E,C) \to (E,(C+\ker(F))/\ker(F)) + (E,(C+\ker(V))/\ker(V)).
$$

What's wrong with the argument in case B?