The quick answer to this question is "depends". It depends on what kind of solution this is, how critical it is for the business, how much is the business willing to spend to maintain the solution, etc.

I'd say there are 2 major components to consider and evaluate: Complexity and Criticality.

If the solution is highly complex and highly critical to the business, then it's probably best to manage it internally because (1) it will most likely require a complex integration process and (2) you want to minimize the number of external dependencies from critical processes.

On the flip side, solutions of low complexity and low criticality are great candidates to be hosted externally because (1) if the external host experiences problems, there will not be a critical impact on the business (2) integration should simple and (3) the overall cost should be lower than an in-house application (cost is not only hardware and software, but also management and maintenance must be considered).

The other 2 scenarios left (high complexity/low criticality and low complexity/high criticality) are very subjective to the type of solution you are trying to implement.

Hope that helps!