What's the best way to have different parent field initialization?

I have a class as below

public abstract class MyObjectManager {
   private final Map<MyKey, MyObject> objects;
   private final MySystem system;
   MyObjectManager(MySystem inputSystem) {
     system = inputSystem;
     // also initialize the "objects" field.
   } 
   public void write(MyKey myKey, MyObject myObject) {...}
   public MyObject read(MyKey myKey) {...}
   public abstract MyObject getNewestObject();
}

I need two types of ConcreteManagers which will have different map implementation, for example,

One uses new LinkedHashMap(CAPACITY_1, LOAD_FACTOR_1, true){ // override the removeEldestEntry(){// logic 1}}.

The other uses new LinkedHashMap(CAPACITY_2, LOAD_FACTOR_2, false){ // override the removeEldestEntry(){// logic 2}}.

I don't want to pass the map as a @param since the map implementation is fixed for each ConcreteManager.

Should I use a strategy pattern or factory pattern to have different map initialization?

Or should I move the objects field to each implementation classes? But their implementation of read and write methods duplicate a lot.