This is a frequent concern. It is in everyone’s best interest, however, that HR does a thorough, unbiased, and neutral investigation, and it is HR’s role to do so. There could be potential legal jeopardy if they do not.
If HR favoritism is what you’re finding, however, then you have some options, but you need to be specific and detailed regarding your allegations of HR bias in investigations. Sorry, but basing it on a “hunch” won’t work. Instead, what specific behaviors or evidence are you basing your claims of bias on? Is the HR investigator a personal friend/social media friend/relative of the complainant? Was the investigation result predetermined without you giving your side (and how do you know that it was predetermined?) Did HR falsify documents?
You can make your complaint at any time
1) before your investigation starts (if there is a glaring conflict of interest, for example)
2) during your investigation (based on intolerable and unprofessional behavior of the investigator), or
3) after your HR investigator has wrapped up his or her investigation.
(Note that doing so after the complaint is decided, however, may make you look like you just didn’t like the result.)
Use your company’s procedures for filing a formal complaint. First take a look at your company’s HR policies to see if they have language that says they will provide a neutral, unbiased investigation (what they were supposed to give you). Reference your HR representative as having violated that policy in your complaint. Then lay out how they violated it.
Typical alternatives for filing a complaint include filing the complaint with your HR investigator’s manager, director, or VP; contacting the employee compliance/complaint line (if your company has one), and complaining through your management chain. Consider doing so in writing so you can document your complaint. HR tends to protect their own. Make sure you indicate that you want to file a formal complaint rather than simply complain.