  In this paper, we consider a charged massive fermionic quantum field in the
space-time of an idealized cosmic string, in the presence of a magnetic field
confined in a cylindrical tube of finite radius. Three distinct configurations
for the magnetic field is taken into account: (i) a cylindrical shell of radius
$a$, (ii) a magnetic field proportional to $1/r$ and (iii) a constant magnetic
field. In these three cases, the axis of the infinitely long tube of radius $a$
coincides with the cosmic string. Our main objective is to analyze the induced
vacuum fermionic current densities outside the tube. In order to do that, we
explicitly construct the wave-functions inside and outside the tube for each
case. Having the complete set of normalized wave-functions, we use the
summation method to develop our analysis. We show that in the region outside
the tube, the induced currents are decomposed into a parts corresponding to a
zero-thickness magnetic flux in addition to a core-induced contributions. The
latter presents specific form depending on the magnetic field configuration
considered. We also see that the only non-vanishing component of fermionic
current corresponds to the azimuthal one. The zero-thickness contribution
depends only on the fractional part of the ration of the magnetic flux inside
the tube by the quantum one. As to the core-induced contribution, it depends on
the total magnetic flux inside the tube, and consequently, in general, it is
not a periodic function of the flux.
