One well-known fact is that putting a ferromagnetic material in an inductor will increase the magnetic flux and thus the inductance value.
Wireless power transfer (WPT) happens through the magnetic near-field. It seems logical that if the magnetic field strength generated by the transmitter is increased so is the field at the receiver. It therefore seems very desirable.
On the other hand all the WPT solutions I have seen use air cores and only a ferromagnetic shield. The high permeability of the shield makes it equivalent to a much thicker air gap, and thus the field at its other end will not be weakened by any metals.
The situation seems to be similar on the receiver side. There are ferrite rod antennas for far-field reception. The connection between two ferrite core coils at a given air gap seems to be the same as if we would take one ferrite rod, wind the two coils on it, and just cut the ferrite rod in the middle and pull it apart a little. The magnetic field should be concentrated in and around the ferrite cores, shouldn't it? Why aren't ferrite cores used in WPT? What are their drawbacks?
PS: an image search has not resulted with any coil using a ferrite core, but the black pad below the planar coil winding is a ferromagnetic material, which concentrates the magnetic flux density. This way the coils can be placed on conductive surfaces, such as the battery of a mobile phone, without great losses.