  In this paper, we analyze the vacuum azimuthal fermionic current induced by a
magnetic field confined in a cylindrical tube of finite radius $a$, in the
cosmic string spacetime. Three distinct configurations for the magnetic field
are 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; moreover, we only develop this analysis for the region outside
the tube. In order to do that, we explicitly construct the corresponding
complete set of normalized wave-functions. We show that in the region outside
the tube, the induced current is decomposed into a part corresponding to a
zero-thickness magnetic flux in addition to a core-induced contribution. The
latter presents specific form depending on the magnetic field configuration
considered. 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.
