Directly from the Latin:
* Augment (verb) - to increase the size or amount of something.
* Augmentation (noun) - the act of increasing something.
Derived from the Latin:
* Augury (noun) - a sign or omen, especially one considered to be a prediction of the future.
* August (adjective) - impressive, majestic, or grand.
* Augur (noun) - a prophet or soothsayer.
* Inaugurate (verb) - to formally begin or introduce something.
* Inauguration (noun) - a ceremony that marks the beginning of something.
Other:
* Augur (verb) - to predict or foreshadow something.
It's important to note that the spelling of the root can vary slightly depending on the word, but it generally maintains the "aug-" core.