  We study model embeddability, which is a variation of the famous embedding
problem in probability theory, when apart from the requirement that the Markov
matrix is the matrix exponential of a rate matrix, we additionally ask that the
rate matrix follows the model structure. We provide a characterisation of model
embeddable Markov matrices corresponding to symmetric group-based phylogenetic
models. In particular, we provide necessary and sufficient conditions in terms
of the eigenvalues of symmetric group-based matrices. To showcase our main
result on model embeddability, we provide an application to hachimoji models,
which are eight-state models for synthetic DNA. Moreover, our main result on
model embeddability enables us to compute the volume of the set of model
embeddable Markov matrices relative to the volume of other relevant sets of
Markov matrices within the model.
