  It is well-known that a point $T\in cv_N$ in the (unprojectivized)
Culler-Vogtmann Outer space $cv_N$ is uniquely determined by its
\emph{translation length function} $||.||_T:F_N\to\mathbb R$. A subset $S$ of a
free group $F_N$ is called \emph{spectrally rigid} if, whenever $T,T'\in cv_N$
are such that $||g||_T=||g||_{T'}$ for every $g\in S$ then $T=T'$ in $cv_N$. By
contrast to the similar questions for the Teichm\"uller space, it is known that
for $N\ge 2$ there does not exist a finite spectrally rigid subset of $F_N$.
  In this paper we prove that for $N\ge 3$ if $H\le Aut(F_N)$ is a subgroup
that projects to an infinite normal subgroup in $Out(F_N)$ then the $H$-orbit
of an arbitrary nontrivial element $g\in F_N$ is spectrally rigid. We also
establish a similar statement for $F_2=F(a,b)$, provided that $g\in F_2$ is not
conjugate to a power of $[a,b]$.
  We also include an appended corrigendum which gives a corrected proof of
Lemma 5.1 about the existence of a fully irreducible element in an infinite
normal subgroup of of $Out(F_N)$. Our original proof of Lemma 5.1 relied on a
subgroup classification result of Handel-Mosher, originally stated by
Handel-Mosher for arbitrary subgroups $H\le Out(F_N)$. After our paper was
published, it turned out that the proof of the Handel-Mosher subgroup
classification theorem needs the assumption that $H$ be finitely generated. The
corrigendum provides an alternative proof of Lemma~5.1 which uses the
corrected, finitely generated, version of the Handel-Mosher theorem and relies
on the 0-acylindricity of the action of $Out(F_N)$ on the free factor complex
(due to Bestvina-Mann-Reynolds). A proof of 0-acylindricity is included in the
corrigendum.
