  We investigate $2n$-dimensional axion electrodynamics for the purpose of
exploring a higher-group structure underlying it. This is manifested as a
Green-Schwarz transformation of the background gauge fields that couple
minimally to the conserved currents. The $n=3$ case is studied most
intensively. We derive the identities of correlation functions among the global
symmetry generators by using a gauge transformation that maps two correlation
functions with each other. A key ingredient in this computation is given by the
Green-Schwarz transformation and the 't Hooft anomalies associated with the
gauge transformation. The algebraic structure of these results and its physical
interpretations are discussed in detail. In particular, we find that the
higher-group structure for $n=3$ is endowed with a multi-ary operation among
the symmetry generators.
