  A natural problem in combinatorial rigidity theory concerns the determination
of the rigidity or flexibility of bar-joint frameworks in $\mathbb{R}^d$ that
admit some non-trivial symmetry. When $d=2$ there is a large literature on this
topic. In particular, it is typical to quotient the symmetric graph by the
group and analyse the rigidity of symmetric, but otherwise generic frameworks,
using the combinatorial structure of the appropriate group-labelled quotient
graph. However, mirroring the situation for generic rigidity, little is known
combinatorially when $d\geq 3$. Nevertheless in the periodic case, a key result
of Borcea and Streinu characterises when a quotient graph can be lifted to a
rigid periodic framework in $\mathbb{R}^d$. We develop an analogous theory for
symmetric frameworks in $\mathbb{R}^d$. The results obtained apply to all
finite and infinite 2-dimensional point groups, and then in arbitrary dimension
they concern a wide range of infinite point groups, sufficiently large finite
groups and groups containing translations and rotations. For the case of finite
groups we also derive results concerning the probability of assigning group
labels to a quotient graph so that the resulting lift is symmetrically rigid in
$\mathbb{R}^d$.
