A power transmission system transmits power from an internal combustion engine to a first and a second auxiliary device through an endless transmitting member. Prior to issuing of an engine start request, the first auxiliary device is actuated to move a tensioner pulley to a given position to increase the degree of tension of the endless transmitting member. The actuation of the first auxiliary device as an engine starter upon the issuing of the engine start request, therefore, enables a drive shaft of the engine to be rotated immediately within a required time to crank the engine. This achieves a quick start of the engine and results in a decrease in amount of time between the issuing of the engine start request and the actual start of the engine.