Specific heat capacity

Does the specific heat capacity of an object affect the rate of transfer of energy from an object to another?
For instance, alcohol has a lower specific heat capacity than water.
If they are both at 50 degree Celsius and poured into a beaker containing water of 20 degree Celsius respectively, after 1 second, in which beaker, much heat is transferred?

Thanks for kind attention.
The specific heat of an item tells you how much energy is required to increase 1.0 grams of the item by 1 degree celsius/kelvin. I would imagine the rate of which the temperature changes would be governed more by the thermodynaic properties of the material, such as its heat conductivity.
Are there any examples showing that the rate does not necessarily depend on the specific heat capacity?

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