Modern Vespa

I purchased a 2009 S50 4 valve with only 75 miles, near mint condition except it sat in a garage with fuel in the tank for years. I am new to working on Vespas and figured an older model would be easier to learn the mechanic basics.

While cleaning out the air filter and spark plug, i noticed a yellow hose attached to a part in the top front of the engine compartment that is not described in the service manual. It also appears that the hose is in need of replacement and was at some point clamped to what think was a nipple under the "auto choke (?)" on the carburetor. Does anyone know what this hose is attached to, what the hose is for and if it actually needs to be reclamped to the carburetor?

Here it the pic of the part that the hose is attached to. Not sure what this is.

Here is the real pic, think I got it right now. Not sure what this hose is attached to.

It's a solenoid. We sell a lot of them. I have no idea what it does. But it must be important.