to suppose  - suponer, imaginarse, creer 

to assume  - suponer, asumir, encargarse de, adquirir, adoptar, tomar, afectar, simular 

to imply  - insinuar, dar a entender, implicar, suponer 

to suggest  - sugerir, indicar, dar a entender 

to involve  - ocupar (con una tarea, etc.), involucrar, enredar, implicar, concernir, afectar, conectar, relacionar, suponer, incluir, consistir en 

to entail  - implicar, suponer, conllevar