The latest word on this issue seems to be The Impact of the Revised Sunspot Record on Solar Irradiance Reconstructions Solar Physics (2016):
Climate sensitivity is uncertain by a factor of
about two and may differ for different forcings, but it is generally considered to be in the
range of 0.2 to 1 °C per [watt per square meter] forcing. The response time for the climate system to reach
equilibrium is also uncertain. This depends primarily on the amount of energy transported
through the near-surface ocean mixed layer to the deep ocean.
An increase in TSI [total solar irradiance] of 1 [watt per sq. m.] produces solar forcing [Delta]Fsol = 0.7×1/4 = 0.18 [watt per sq. m.],
where the scale factors account for the Earth’s albedo and the geometric illumination of the
entire Earth surface by the Sun. Assuming a climate sensitivity to solar forcing of 0.6 °C per
[watt per sq. m.] (i.e. midway in the estimated range), an increase in TSI of about 1 [watt per sq. m.] from the seventeenth century Maunder Minimum to the present solar-cycle average produces global
warming of 0.1 °C. This estimated warming is reduced by about 20 % to 0.08 °C in the
NRLTSI2 model and is not reduced at all in the SATIRE model when using the new sunspot
record to reconstruct TSI. The difference is small and within the uncertainty of the TSI reconstructions
themselves (Figures 5 and 6). Furthermore, this possible solar-caused global
surface-temperature increase is significantly lower than the net measured temperature increase
of at least 0.8 °C over the same four-century time frame.
The net effect on climate according to this new sunspot-number record is at a nearly
insignificant level. Using the SILSO record, global-warming estimates that are attributable
to solar variability over the last four centuries may be up to 20%lower than current estimates
suggest, perhaps causing only a 0.08 °C increase instead of a possible 0.1 °C increase in
global surface temperature. These differences are negligible compared to the uncertainties,
and either contribution to overall global warming remains much lower than those that are
due to other climate influences.
So half of 0.6 °C being due to change in the Sun's output is ruled out.
On the other hand, even more recently, there is Solar activity has a direct impact on Earth's cloud cover and the corresponding journal article The response of clouds and aerosols to cosmic ray decreases Journal of Geophysical Research Volume 121, September 2016, Pages 8152–8181
which, quoting from the popular article, says:
"Earth is under constant bombardment by particles from space called galactic cosmic rays. Violent eruptions at the Sun's surface can blow these cosmic rays away from Earth for about a week. Our study has shown that when the cosmic rays are reduced in this way there is a corresponding reduction in Earth's cloud cover. Since clouds are an important factor in controlling the temperature on Earth our results may have implications for climate change", explains lead author on the study Jacob Svensmark of DTU
So overall the direct effect of total solar irradiance on the climate over the past 110 years has been shown to be small, but indirect effects, such as regulation of cloud formation by controlling incoming cosmic rays, have not been ruled out.