  This paper is a continuation of papers
\cite{UmarovTsallisSteinberg,UmarovTsallisGellmannSteinberg}. In Part I
\cite{UmarovTsallisGellmannSteinberg} a description (representation) of
$(q,\alpha)$-stable distributions based on a $F_q$-transform was given. Here,
in Part II, we present another description of these distributions. This
approach generalizes results of \cite{UmarovTsallisSteinberg} (which
corresponds to $\alpha=2, Q\in [1,3)$) to the whole range of stability and
nonextensivity parameters $\alpha \in (0,2]$ and $Q \in [1,3),$ respectively.
The present case $\alpha=2$ recovers the $q$-Gaussian distributions. Similar to
what is discussed in \cite{UmarovTsallisSteinberg}, a triplet
$(q^{\ast},q,q_{\ast})$ arises for which the mapping $F_{q^{\ast}}:
\mathcal{G}_{q} \to \mathcal{G}_{q_{\ast}}$ holds. Moreover, by unifying the
two preceding descriptions, further possible extensions are discussed and some
conjectures are formulated.
