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We investigate the interactions of in-phase Airy beams modulated by a fundamental Gaussian beam and fourth-order diffraction in Kerr nonlinear media. Directly numerical simulations show that normal (anomalous) fourth-order diffraction and an in-phase (out-of-phase) Gaussian beam affect the interactions of solitons generated from Airy beams in unique ways. Different from previous results, suggesting that interactions of in-phase (out-of-phase) conventional beams are always attractive (repulsive), many anomalous interactions of Airy beams are obtained. Stable breathing Airy soliton pairs can be formed with fourth-order diffraction and a fundamental Gaussian beam.