Frying with Induction Stovetop
Induction just heats the pan using magnetism. Aside from very high efficiency, it isn't so different from any other method, and how hot the oil gets is an issue of power output and heat transfer/conductivity of the pan itself. There isn't an appreciable difference in how the physics work inside the pan.

Clearly your oil is not hot enough for frying/sautéeing. Most induction cooktops say they draw at least 1000 W to produce temperatures in the 350-400° F range, which is a somewhat typical frying temperature, being just at or below the smoke points of most oils. However, that number is based on a lot of assumptions, and 1500 W is not necessarily a generous margin of error for a portable hot plate. A lot of things could prevent you from reaching even a modest 400° F:


The pan is too dense. Are you using cast iron? It responds well to induction but the thermal resistance is also a lot higher.
The pan is very large and/or you're stuffing a lot of food into it, a result of which could be that the heat is escaping faster than the hot plate can pump it out.
The pan does not actually have the right amount of magnetic permeability for high-heat cooking. A magnet may stick to it, but how hard does it stick? If there isn't a very strong attraction - i.e. you have to put some effort into pulling off a permanent magnet - then you're not getting anywhere near the efficiency the unit was designed for, and even the full 1500 W may not be enough to reach the desired temperatures.
The cooktop is not really pulling in the full 1500 W, either due to a poor power source or simply poor design. Some manufacturers might quote 1500 W as a peak in order to sell more units, knowing full well that they can't sustain more than about 60% of that. If you've got a power meter (e.g. Kill-A-Watt), plug it in and see how much it's really drawing.


There are some portable induction cookers that go up to 1800 or even 2000 W, which is the bare minimum for the large element on an electric or induction stove (2500 W is typical for those). If you want to use a portable hot plate for high-heat cooking, check your pans very carefully, don't overcrowd them, and if all else fails, you might just need a better hot plate.