The investigator extract Defensive Grace says “each round you can roll an inspiration die without expending a use of inspiration and gain the result as a dodge bonus to AC for that round.“ This is a personal extract, however, with infusion the investigator can give it to others who do not have inspiration. Can they gain the benefits of this extract?
If so, do they gain the use of the inspiration die from the investigator who gave them the infusion?
The extract does other things - makes the drinker immune to precision damage (sneak attack, etc.) and can be dismissed to turn a critical hit against you into a normal hit. Those definitely work as an imbued extract.
As for the inspiration die, I have to think this falls into the "something the writer didn't consider" category. Which means there's no "correct" answer. In the absence of a clear reason why it would not work, my ruling is that the imbued extract WOULD allow the drinker to gain the inspiration die. The die would be 1d6, the base for inspiration, unless the drinker had a higher inspiration die himself (another investigator). Spells and abilities that allow the recipient to use the caster's attributes specifically say so. For example, the imbue hex spell on the same page of ACO says that the transferred hex uses the caster's level, not the recipient.