  In this paper we extend a previous investigation by us regarding an iterative
construction of irreducible polynomials over finite fields of odd
characteristic. In particular, we show how it is possible to iteratively
construct irreducible polynomials by means of two families of transforms, which
we call the $Q_k$ and $\hat{Q}_k$-transforms, related to certain degree two
isogenies over elliptic curves, which split the multiplication-by-$2$ map.
