According to this list, this particular group is described as:

> The Base is a predominantly US based militant white supremacist group, formed in 2018. It draws influence from a collection of essays by prominent National Socialist, James Mason, which advocate the use of violence to initiate the collapse of modern society through a ‘race war’ and the subsequent creation of a white ethno-state. This ideology is known as ‘accelerationism’.

Al Qa’ida (AQ) are listed separately:

> Inspired and led by Usama Bin Laden, its aims are the expulsion of Western forces from Saudi Arabia, the destruction of Israel and the end of Western influence in the Muslim world.

No, although the name was likely picked to cause exactly such confusion. 'Al Qaeda' is commonly translated as 'the base' but there's no connection between the two. AQ was a pretty elaborate and complex organization in its heyday though it seems mostly moribund now. The Base existed mostly virtually and was more focused on promoting stochastic terrorism by disseminating incitement and instructional materials, but without any real infrastructure/finance.