  We characterize the subsets $\Gamma$ of $\C$ for which the notion of
$\Gamma$-supercyclicity coincides with the notion of hypercyclicity, where an
operator $T$ on a Banach space $X$ is said to be $\Gamma$-supercyclic if there
exists $x\in X$ such that $\overline{\text{Orb}}(\Gamma x, T)=X$. In addition
we characterize the sets $\Gamma \subset \C$ for which, for every operator $T$
on $X$, $T$ is hypercyclic if and only if there exists a vector $x\in X$ such
that the set $\text{Orb}(\Gamma x, T)$ is somewhere dense in $X$. This extends
results by Le\'on-M\"uller and Bourdon-Feldman respectively. We are also
interested in the description of those sets $\Gamma \subset \C$ for which
$\Gamma$-supercyclicity is equivalent to supercyclicity.
