Work in ZF. Let $\lambda$ be strongly inaccessible. Let $\kappa$ be such that for every $\alpha\lt\lambda$, there is some $j: V_\lambda\prec V_\lambda$, with critical point $\kappa$, such that $j(\kappa)\gt\alpha$. Then, can we define some set $A=\{j_\alpha: V_\lambda\prec V_\lambda|\alpha\lt\lambda\}$, such that every $j_\alpha$ is a non-trivial elementary embedding with critical point $\kappa$, and $j(\kappa)\gt\alpha$?

(Note: $A$ needs to be a sequence, not just the set of all embeddings)

Motivation: I was doing research on large cardinals without choice, and was attempting to use a similar set of embedding to $A$ to give a first order definition of cardinals that were $n$-huge.