EX-10.5





 Exhibit 10.5 

INVESTINDUSTRIAL ACQUISITION CORP. 

Suite 1, 3rd Floor, 11-12 St James’ Square 

London SW1Y 4LB 
 United Kingdom

 [●], 2020 
 Investindustrial
Acquisition Corp. L.P. 
 Suite 1, 3rd Floor, 11-12 St James’ Square 

London SW1Y 4LB 
 United Kingdom 

Ladies and Gentlemen: 
 This letter will confirm
our agreement that, commencing on the effective date (the “Effective Date”) of the registration statement (the “Registration Statement”) for the initial public offering (the
“IPO”) of the securities of Investindustrial Acquisition Corp. (the “Company”) and continuing until the earlier of (i) the consummation by the Company of an initial business combination and
(ii) the Company’s liquidation (in each case as described in the Registration Statement) (such earlier date hereinafter referred to as the “Termination Date”), Investindustrial Acquisition Corp. L.P. (the
“Sponsor”) shall take steps directly or indirectly to make available to the Company certain office space, secretarial and administrative services as may be required by the Company from time to time, situated at Suite 1, 3rd
Floor, 11-12 St James’ Square, London SW1Y 4LB, United Kingdom (or any successor location). In exchange therefore, the Company shall pay the Sponsor a sum of up to $10,000 per month on the Effective Date
and continuing monthly thereafter until the Termination Date. The Sponsor hereby agrees that it does not have any right, title, interest or claim of any kind (a “Claim”) in or to any monies that may be set aside in a trust
account (the “Trust Account”) that may be established upon the consummation of the IPO and hereby irrevocably waives any Claim it may have in the future as a result of, or arising out of, any negotiations, contracts or
agreements with the Company and will not seek recourse against the Trust Account for any reason whatsoever. 
 This letter agreement
constitutes the entire agreement and understanding of the parties hereto in respect of its subject matter and supersedes all prior understandings, agreements, or representations by or among the parties hereto, written or oral, to the extent they
relate in any way to the subject matter hereof or the transactions contemplated hereby. 
 This letter agreement may not be amended,
modified or waived as to any particular provision, except by a written instrument executed by the parties hereto. 
 The parties may not
assign this letter agreement and any of their rights, interests, or obligations hereunder without the consent of the other party. 
 This
letter agreement shall be governed by, construed in accordance with, and interpreted pursuant to the laws of the State of New York, without giving effect to its choice of laws principles that will apply the laws of another jurisdiction. 

This letter agreement may be executed in one or more counterparts, each of which shall for all purposes be deemed to be an original but all of
which together shall constitute one and the same agreement. Only one such counterpart signed by the party against whom enforceability is sought needs to be produced to evidence the existence of this letter agreement. 

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	Very truly yours,
	
	INVESTINDUSTRIAL ACQUISITION CORP.
		
	By:	 	
                     
    

	Name:	 	
	Title:	 	


 AGREED TO AND ACCEPTED BY: 

INVESTINDUSTRIAL ACQUISITION CORP. L.P. 
 ACTING BY ITS
GENERAL PARTNER, 
 ACQUISITION CORP. GP LIMITED 
  

			
	By:	 	  

	Name:	 	
	Title:	 	
		
	By:	 	  

	Name:	 	
	Title: