An investigation of the electron impact ionization and fragmentation of helium clusters that contain Ne atoms and Ne-k subclusters has been performed. The charge transfer probability from He+ to Ne and the branching ratios for fragmentation of the Ne-k subclusters were found by analyzing the dependence of the ion signal intensities on the Ne pressure in the "pickup＇＇ region. The measured charge transfer probability from He+ to Ne ranges from 0.06+/-0.01 for clusters of mean original size [N] = 3300 to 0.43 +/- 0.02 for [N] = 1100, Charge transfer to a single Ne atom within the helium clusters never yields bare Ne+ ions. Instead, fragments of the type NeHen+ are produced. The charge transfer from He+ to Ne-2 subclusters yields mainly Ne-2(+) for smaller initial cluster sizes, but NeHen+ or Ne2Hen+ fragments are more probable for larger clusters. This shows that He droplets of a few thousand atoms are able to cage Ne-2 subclusters by dissipating the entire energy released by charge transfer and formation and vibrational relaxation of the Ne-2(+) ion. Interestingly, it was found that in these relatively small helium clusters the Ne-3 and Ne-4 subclusters never survive the charge transfer from He+. Fragments such as Ne-2(+) and Ne2Hen+ are more likely to survive than are Ne-3(+) and Ne-4(+). In general, the results presented here are qualitatively similar to those for a recent study of the ionization of Ar in helium droplets. In both cases fragmentation to the bare ion is rare, while fragmentation to the dimer ion dominates. However, the helium cluster caging effect is more efficient for Ne subclusters than for Ar subclusters. Also, there is no evidence for shell structures in the NeHen+ ion fragment distributions.