  We ascertain properties of the algebraic structures in towers of codes,
lattices, and vertex operator algebras (VOAs) by studying the associated
subobjects fixed by lifts of code automorphisms. In the case of sublattices
fixed by subgroups of code automorphisms, we identify replicable functions that
occur as quotients of the associated theta functions by suitable eta products.
We show that these lattice theta quotients can produce replicable functions not
associated to any individual automorphisms. Moreover, we show that the
structure of the fixed subcode can induce certain replicable lattice theta
quotients and we provide a general code theoretic characterization of order
doubling for lifts of code automorphisms to the lattice-VOA. Finally, we prove
results on the decompositions of characters of fixed subVOAs.
