  If we take a (bar-joint) framework, prepare an identical copy of this
framework, translate it by some vector $\tau$, and finally join corresponding
points of the two copies, then we obtain a framework with `extrusion' symmetry
in the direction of $\tau$. This process may be repeated $t$ times to obtain a
framework whose underlying graph has $\mathbb{Z}_2^t$ as a subgroup of its
automorphism group and which has `$t$-fold extrusion' symmetry. We show that
while $t$-fold extrusion symmetry is not a point-group symmetry, the rigidity
matrix of a framework with $t$-fold extrusion symmetry can still be transformed
into a block-decomposed form in the analogous way as for point-group symmetric
frameworks. This allows us to use Fowler-Guest-type character counts to analyse
the mobility of such frameworks. We show that this entire theory also extends
to the more general point-hyperplane frameworks with $t$-fold extrusion
symmetry. Moreover, we show that under suitable regularity conditions the
infinitesimal flexes we detect with our symmetry-adapted counts extend to
finite (continuous) motions. Finally, we establish an algorithm that checks for
finite motions via linearly displacing framework points along velocity vectors
of infinitesimal motions.
