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We study higher-form global symmetries and a higher-group structure of alow-energy limit of $(3+1)$-dimensional axion electrodynamics in a gapped phasedescribed by a topological action. We argue that the higher-form symmetriesshould have a semi-strict 4-group (3-crossed module) structure by consistencyconditions of couplings of the topological action to background gauge fieldsfor the higher-form symmetries. We find possible 't Hooft anomalies for the4-group global symmetry, and discuss physical consequences.

Global 4-group symmetry and 't Hooft anomalies in topological axion electrodynamics