The HI extension of the standard HI spaces

A Hereditarily Indecomposable (HI) Banach space $X$ admits an HI extension if there exists an HI space $Z$ such that $X$ is isomorphic to a subspace $Y$ of $Z$ and $Z/Y$ is of infinite dimension. The problem whether or not every HI space admits an HI extension is attributed to A. Pelczynski. In this paper we present a method to define HI-extensions of the standard HI spaces, a class which includes the Gowers-Maurey space, asymptotic $\ell_{p}$-HI spaces and others.