A silicone rubber composition having such selective adhesiveness that shows high adhesion to an organic material than to a molding die is bonded to a sheet of the organic material and is vulcanized and cured in the mold. Thus, an elastomer keypad of double-layer structure is produced in which an organic material layer, which can be easily subjected to coating, bonding with an adhesive or pressure-sensitive adhesive, or the like, has been united and bonded as a surface layer to the rubber. Keytops are bonded to the elastomer keypad to produce a covering member for push-button switches. A polyurethane elastomer sheet is preferably used as the organic material.