Extension operators are used in various applications, such as fluid-structure interaction or shape optimization with the method of mappings, to map a mesh of a reference domain to a deformation of the domain, by extending boundary deformations to the interior. Typically, extension operators are defined by the solution of PDEs with Dirichlet boundary conditions, but PDE-based extension operators that can handle larger deformations can be prohibitively computationally expensive. Therefore, we investigate replacing these operators with learned neural network surrogates. Taking inspiration from the PDE nature of typical extension operators and the graph structure of meshes, we propose operator learning- and graph neural network-based methods tailored for the specific application of learning extension operators. We discuss aspects of network architecture, training and implementation and evaluate them as components in finite element simulations.