No, you can't share your own extra attacks with your beast companion
Looking at the Beast Master features, we see the following:
You can use your action to verbally command it to take the Attack, Dash, Disengage, or Help action. If you don't issue a command, the beast takes the Dodge action. Once you have the Extra Attack feature, you can make one weapon attack yourself when you command the beast to take the Attack action. (PHB, Chapter 3: Classes, Ranger, Beast Master, Ranger's Companion)
This section doesn't state that you can allow your companion to freely make use of your own multiple attacks, it only says that you can direct the companion to attack and also attack once yourself.
Additionally, there is a class feature which allows your companion to attack more than once, and importantly it even states that the companion can't make multiple attacks using its own multiattack feature (if it has one) until that class feature is gained:
Starting at 11th level, when you command your beast companion to take the Attack action, the beast can make two attacks, or it can take the Multiattack action if it has that action. (PHB, Chapter 3: Classes, Ranger, Beast Master, Ranger's Companion, Bestial Fury)
The companion's ability to attack is a function of its Beast Master's levels in the Ranger class, not the Beast Master's other class levels, abilities, nor even the companion's own stat block.
That said, if you wanted to homebrew something like that option, it probably wouldn't be too hard to integrate into the game.