Greenshoe - Issuer or underwriter's option?

I have read about this and am bit confused. Is this an option the issuer or the underwriter has? I also understand this is an option that can be exercised at listing or within 25 days.

Example: The stock rises after listing.

1. If the issuer has the greenshoe option and the price rises after IPO: Would want to sell more shares to the public and buys these shares from the issuer at lower price. Books a profit. The company raises more capital and the amount of outstanding shares increase.

2. If issuer has the greenshoe option and the price rises after IPO: Would not want to issue more shares at the IPO price, when the share price is already higher.

Is my understanding correct? 

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