An apparatus deploys and retrieves a wear bushing in a wellhead during the same trip, such as when using a bottom hole assembly for drilling. Arms in the form of pistons, levers, dogs, etc. can be moved between retracted and extended conditions on the apparatus using actuations from remote communications. When extended, the arms engage inside the wear bushing or inside an adapter disposed in the bushing. When retracted, the arms lie within the exterior of the tool to avoid wear and damage when the drillstring is used for drilling or other operations. During deployment, the bushing supported on the extended arms is seated in the wellhead. The arms are then retracted through remote activation, and the bottom hole assembly is run further downhole. During retrieval, the retracted arms are extended through remote activation and engage in the bushing so the drillstring can lift the bushing from the wellhead.