Nonlinear processes of laser beams carrying orbital angular momentum (OAM) offer a means to generate new wavelengths and to manipulate OAM charge numbers. We demonstrate the second-harmonic generation (SHG) of asymmetric Bessel-Gaussian (BG) beams carrying OAM of both integer and fractional charge numbers. Experimental results show a good one-to-one correspondence of the charge numbers and compliance with the OAM conservation law. The SHG conversion process and efficiency with different combined charge numbers are also discussed.