A:
Include basically means to add on or incorporate into something. Examples:
"Does this price include sales tax?"
"Shipping and taxes are not included."
"The ceremony will include a presentation by David and Susan."
"His duties include washing dishes, cleaning the kitchen, and setting the table."
"Please include your name and phone number on the report."
"This house has 8 rooms if you include the entrance hall."
"Is breakfast included?" (asking if you get a free breakfast when you get a hotel room)
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Involve basically refers to something that is required or someone who is actively participating in something (like an event, conversation, plan, activity, etc.).
"Are you involved in politics?"
"His job involves a lot of hard work."
"Should we involve Kevin in the conversation?" ***
"Please do not involve him in our conversation." ***
"Susan often gets involved in conflicts and arguments."
"He is involved in a lot of criminal activity."
"You should not get involved with him." (could suggest a romantic relationship)
"The report I am writing involves a lot of research."
*** = Both "include" and "involve" are acceptable for these sentences.