The invention relates to processes for the production of a composite component via multicomponent injection molding, where the composite component comprises a main body composed of a thermoplastic and an external layer composed of a foamed thermoplastic. The main body is produced via injection molding and subsequent hardening of the thermoplastic. A mixture of the foamed thermoplastic and a chemical blowing agent is injected onto the main body, subsequently hardened in the same injection mold, and foamed by heating. The composite component is solidified via cooling. The foamed thermoplastic is at least one polyamide whose modulus of elasticity is from 1 MPa to 250 MPa in the entire temperature range from 70° C. to 180° C. The chemical blowing agent is in non-polymerized form.