  Let $L$ be a linear set of pseudoregulus type in a line $\ell$ in
$\Sigma^*=\mathrm{PG}(t-1,q^t)$, $t=5$ or $t>6$. We provide examples of
$q$-order canonical subgeometries $\Sigma_1,\, \Sigma_2 \subset \Sigma^*$ such
that there is a $(t-3)$-space $\Gamma \subset \Sigma^*\setminus (\Sigma_1 \cup
\Sigma_2 \cup \ell)$ with the property that for $i=1,2$, $L$ is the projection
of $\Sigma_i$ from center $\Gamma$ and there exists no collineation $\phi$ of
$\Sigma^*$ such that $\Gamma^{\phi}=\Gamma$ and $\Sigma_1^{\phi}=\Sigma_2$.
  Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des. Codes
Cryptogr. 56:89-104, 2010) states the existence of a collineation between the
projecting configurations (each of them consisting of a center and a
subgeometry), which give rise by means of projections to two linear sets. It
follows from our examples that this condition is not necessary for the
equivalence of two linear sets as stated there. We characterize the linear sets
for which the condition above is actually necessary.
