In the context of Bayesian inversion, we consider sequential Monte Carlo (SMC) methods that provide an approximation of the posterior probability density function and the evidence (marginal likelihood). These particle approaches build a sequence of importance sampling steps between gradually tempered distributions evolving from the prior to the posterior PDF. To automate the definition of the tempering schedule, adaptive SMC (ASMC) allows tuning the temperature increments on-the-go. One general challenge in Bayesian inversions is the computational burden associated with expensive, high-fidelity forward solvers. Lower-fidelity surrogate models are interesting in this context as they can emulate the response of expensive forward solvers at a fraction of their cost. We consider surrogate modelling within ASMC and introduce first an approach involving surrogate modelling only, in which either prior samples are used to train the surrogate, or the surrogate model is retrained by updating the training set during the inversion. In our implementation, we rely on polynomial chaos expansions for surrogate modelling, principal component analysis for model parametrization and a ground-penetrating radar cross-hole tomography problem with either an eikonal or finite-difference time-domain solver as high-fidelity solver. We find that the method based on retraining the surrogate during the inversion outperforms the results obtained when only considering prior samples. We then introduce a computationally more expensive multifidelity approach including a transition to the high-fidelity forward solver at the end of the surrogate-based ASMC run leading to even more accurate results. Both methods result in speed-ups that are larger than one order of magnitude compared to standard high-fidelity ASMC inversion.