Mac OS X Lion does not update chrome or I can install any application

Chrome is no longer updated for Lion. In fact you need a minimum of 10.9, Mavericks to run a current version of Chrome.

Older versions of Chrome are found online, but their security is mixed. If your Mac is too old to run anything newer than Lion,

you might want to try Tenfourfox, or Firefox Legacy https://parrotgeek.com/fxlegacy.html

Go to Apple menu -> About this Mac, System Profile or Info. In it you will find the machine ID for your Mac as indicated in the brackets below. These are all machines that can upgraded to El Capitan for more recent more secure web browsing:

• MacBook (Early 2015) [8,1]
• MacBook (Late 2008 Aluminum, or Early 2009 or newer) [5,1 or 5,2]
• MacBook Pro (Mid/Late 2007 or newer) [3,1]
• MacBook Air (Late 2008 or newer) [2,1]
• Mac mini (Early 2009 or newer) [3,1]
• iMac (Mid 2007 or newer) [7,1]
• Mac Pro (Early 2008 or newer) [3,1] *
• Xserve (Early 2009) [3,1]

* Note: As the model #s are very similar to 10.8, 10.9, and 10.10, it is possible that the solution that allows earlier Mac Pros to be used may be feasible for 10.11.