I am aware that alpha particle can induce radioactivity. Is it also true for beta and gamma rays?
Yes. For example, a sufficiently energetic beta or gamma can excite a nucleus, which will then deexcite. But these processes all have low probability. As a rule of thumb, secondary radioactivity is never significant except with neutrons.
"Photofission" is the process of gamma rays causing fission reactions.
"Photoneutrons" are when neutrons are produced from (g,n) reactions, but there is no fission. There are not many isotopes that generate photoneutrons, examples include deuterium (H-2) and beryllium (Be-9). See https://www.nuclear-power.net/nuclear-power/fission/delayed-neutrons/photoneutrons/ for more information. Note that the gamma energy must be fairly high.
I am not aware of any beta reactions that create a neutron. The energy of beta particles is usually not high enough. (I'm not ruling it out, I'm just not aware of any.)